Basic

📁 Source: Mathlib/Data/Complex/Basic.lean

Statistics

Metric	Count
Definitions I, Rectangle, instSMulRealComplex, addCommGroup, addGroupWithOne, commRing, equivRealProd, equivRealProdAddHom, im, imAddGroupHom, instAdd, instCoeReal, instCommSemiring, instDecidableEq, instDivInvMonoid, instField, instInhabited, instIntCast, instInv, instMul, instNNRatCast, instNatCast, instNeg, instOne, instRatCast, instRepr, instRing, instSemiring, instStarRing, instSub, instZero, normSq, ofReal, ofRealHom, re, reAddGroupHom, reProdIm, «term_×ℂ_», termℂ	39
Theorems I_im, I_mul, I_mul_I, I_mul_im, I_mul_re, I_ne_zero, I_pow_eq_pow_mod, I_pow_four, I_pow_three, I_re, I_sq, add_conj, add_im, add_re, canLift, coe_imAddGroupHom, coe_reAddGroupHom, conj_I, conj_eq_iff_im, conj_eq_iff_re, conj_eq_iff_real, conj_im, conj_inv, conj_natCast, conj_neg_I, conj_ofNat, conj_ofReal, conj_re, div_I, div_im, div_intCast, div_intCast_im, div_intCast_re, div_natCast, div_natCast_im, div_natCast_re, div_ofNat, div_ofNat_im, div_ofNat_re, div_ofReal, div_ofReal_im, div_ofReal_re, div_ratCast, div_ratCast_im, div_ratCast_re, div_re, equivRealProdAddHom_apply, equivRealProdAddHom_symm_apply, equivRealProdAddHom_symm_apply_im, equivRealProdAddHom_symm_apply_re, equivRealProd_apply, equivRealProd_symm_apply, eta, exists, ext, ext_iff, forall, horizontalSegment_eq, im_eq_sub_conj, im_mul_ofReal, im_nnqsmul, im_nnratCast, im_nsmul, im_ofNat, im_ofReal_mul, im_qsmul, im_sq_le_normSq, im_surjective, im_zsmul, instCharZero, instIsAddTorsionFree, instNontrivial, intCast_im, intCast_re, inv_I, inv_def, inv_im, inv_re, inv_zero, mem_reProdIm, mk_eq_add_mul_I, mul_I_im, mul_I_re, mul_conj, mul_im, mul_inv_cancel, mul_re, natCast_im, natCast_re, neg_im, neg_re, normSq_I, normSq_add, normSq_add_mul_I, normSq_apply, normSq_conj, normSq_div, normSq_eq_conj_mul_self, normSq_eq_zero, normSq_intCast, normSq_inv, normSq_mk, normSq_mul, normSq_natCast, normSq_neg, normSq_nonneg, normSq_ofNat, normSq_ofReal, normSq_one, normSq_pos, normSq_ratCast, normSq_sub, normSq_zero, ofRealHom_eq_coe, ofReal_add, ofReal_comp_add, ofReal_comp_mul, ofReal_comp_neg, ofReal_comp_nsmul, ofReal_comp_pow, ofReal_comp_sub, ofReal_comp_zsmul, ofReal_def, ofReal_div, ofReal_eq_one, ofReal_eq_zero, ofReal_im, ofReal_inj, ofReal_injective, ofReal_intCast, ofReal_inv, ofReal_mul, ofReal_mul', ofReal_natCast, ofReal_ne_one, ofReal_ne_zero, ofReal_neg, ofReal_nnqsmul, ofReal_nnratCast, ofReal_nsmul, ofReal_ofNat, ofReal_one, ofReal_pow, ofReal_qsmul, ofReal_ratCast, ofReal_re, ofReal_sub, ofReal_zero, ofReal_zpow, ofReal_zsmul, one_im, one_re, preimage_equivRealProd_prod, range_im, range_re, ratCast_im, ratCast_re, reProdIm_eq_empty, reProdIm_nonempty, reProdIm_subset_iff, reProdIm_subset_iff', re_add_im, re_eq_add_conj, re_mul_ofReal, re_nnqsmul, re_nnratCast, re_nsmul, re_ofNat, re_ofReal_mul, re_qsmul, re_sq_le_normSq, re_surjective, re_zsmul, real_smul, smul_im, smul_re, star_def, sub_conj, sub_im, sub_re, verticalSegment_eq, zero_im, zero_re	183
Total	222

Complex

Definitions

Name	Category	Theorems
I 📖	CompOp	391 math math: Real.hasDerivAt_fourier, verticalSegment_eq, tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero, log_I, ProbabilityTheory.isGaussian_iff_gaussian_charFun, nnnorm_exp_I_mul_ofReal, log_neg_one, restrictScalars_one_smulRight', GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I, Real.hasFDerivAt_fourierChar_neg_bilinear_left, EisensteinSeries.hasSum_e2Summand_symmetricIcc, mk_eq_add_mul_I, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, HurwitzZeta.hasSum_int_cosKernel₀, exp_eq_exp_iff_exists_int, Real.fourierIntegralInv_eq', cos_mul_I, cos_sub_sin_I, GaussianInt.toComplex_def', integral_boundary_rect_eq_zero_of_differentiableOn, MeasureTheory.charFunDual_apply, ProbabilityTheory.hasGaussianLaw_iff_charFunDual_map_eq, summableLocallyUniformlyOn_iteratedDerivWithin_cexp, equivRealProdAddHom_symm_apply, ModularForm.eta_q_eq_pow, inv_I, HasDerivWithinAt.complexToReal_fderiv', Orientation.kahler_comp_rightAngleRotation', ZMod.LFunction_one_sub, arg_neg_I, Real.fderiv_fourierChar_neg_bilinear_right_apply, exp_add_mul_I, imaginaryPart_I_smul, SchwartzMap.lineDerivOp_fourier_eq, arg_cos_add_sin_mul_I_coe_angle, circleMap_neg_pi_div_two, angle_exp_one, circleIntegral.integral_sub_inv_of_mem_ball, hasSum_one_div_nat_pow_mul_fourier, TemperedDistribution.lineDerivOp_fourier_eq, norm_eq_one_iff', circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, fourierIntegral_gaussian_innerProductSpace', sqrt_I, fourier_gaussian_pi', iteratedDerivWithin_tsum_cexp_eq, jacobiTheta₂_functional_equation, ProbabilityTheory.hasGaussianLaw_iff_charFun_map_eq, sum_cauchyPowerSeries_eq_integral, Real.norm_exp_I_mul_ofReal_sub_one_le, Real.vector_fourierIntegral_eq_integral_exp_smul, HurwitzZeta.evenKernel_def, Real.fourier_real_eq_integral_exp_smul, HurwitzZeta.jacobiTheta₂'_functional_equation', norm_cos_add_sin_mul_I, hasSum_nat_jacobiTheta, Real.fourier_deriv, DirichletCharacter.norm_LSeries_product_ge_one, ZMod.stdAddChar_coe, DifferentiableOn.circleIntegral_sub_inv_smul, HarmonicAt.analyticAt_complex_partial, Real.nnnorm_exp_I_mul_ofReal_sub_one_le, ModularForm.tsum_logDeriv_eta_q, coe_orthonormalBasisOneI, integral_circleTransform, HurwitzZeta.oddKernel_def', isPrimitiveRoot_exp_rat, VectorFourier.fourierPowSMulRight_eq_comp, mem_rootsOfUnity, MeasureTheory.charFun_apply, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, exp_antiperiodic, MeasureTheory.charFun_apply_real, fourierCoeff_bernoulli_eq, exp_add_pi_mul_I, log_exp_eq_re_add_toIocMod, circleMap_eq_circleMap_iff, SchwartzMap.fourier_lineDerivOp_eq, equivRealProd_symm_apply, UpperHalfPlane.re_add_im, TemperedDistribution.fourierInv_lineDerivOp_eq, RCLike.complexRingEquiv_apply, InnerProductSpace.laplacianWithin_eq_iteratedFDerivWithin_complexPlane, arg_cos_add_sin_mul_I, cos_eq_iff_quadratic, cauchyPowerSeries_apply, DiffContOnCl.circleIntegral_sub_inv_smul, ofReal_cpow_of_nonpos, HurwitzZeta.expZeta_one_sub, ProbabilityTheory.complexMGF_mul_I, Real.circleAverage_eq_circleIntegral, norm_exp_I_mul_ofReal_sub_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod, arg_cos_add_sin_mul_I_sub, equivRealProdCLM_symm_apply, cos_eq_tsum', mellinInv_eq_fourierIntegralInv, Real.Angle.coe_toCircle, circleMap_pi_div_two, normSq_I, cos_add_sin_mul_I_pow, mellinInv_eq_fourierInv, norm_eq_one_iff, exp_ofReal_mul_I_re, fourier_gaussian_innerProductSpace', arg_mul_cos_add_sin_mul_I, HurwitzZeta.hasSum_int_cosKernel, sqrt_neg_of_nonneg, exp_neg_pi_div_two_mul_I, AnalyticOnNhd.eq_const_add_im_mul_I_of_re_eq_const, inner_map_polarization', EisensteinSeries.D2_S, Polynomial.int_cyclotomic_rw, EisensteinSeries.G2_S_transform, HurwitzZeta.oddKernel_def, isIntegral_exp_rat_mul_pi_mul_I, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, realPart_add_I_smul_imaginaryPart, HarmonicAt.differentiableAt_complex_partial, Real.fourierIntegral_iteratedDeriv, ProbabilityTheory.IsGaussian.charFun_eq', jacobiTheta_S_smul, I_pow_eq_pow_mod, isPrimitiveRoot_iff, exp_periodic, norm_exp_mul_sq_le, exp_nat_mul_two_pi_mul_I, CliffordAlgebraComplex.ofComplex_I, log_exp_eq_sub_toIocDiv, ProbabilityTheory.complexMGF_id_mul_I, liftAux_neg_I, selfAdjoint.unitarySelfAddISMul_coe, isPrimitiveRoot_exp_of_isCoprime, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, integral_exp_mul_I_eq_sinc, UpperHalfPlane.norm_exp_two_pi_I_lt_one, HurwitzZeta.hasSum_int_cosZeta, arg_exp_mul_I, BoundedContinuousFunction.innerProbChar_apply, sub_conj, EisensteinSeries.hasSum_e2Summand_symmetricIco, ModularForm.eta_q_eq_cexp, HasDerivAt.complexToReal_fderiv', arg_mul_cos_add_sin_mul_I_coe_angle, circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const, EisensteinSeries.q_expansion_riemannZeta, Real.fourierIntegral_real_eq_integral_exp_smul, fourier_coe_apply, Real.deriv_fourierIntegral, liftAux_I, integral_exp_mul_I_eq_sin, cos_add_sin_I, hasSum_cauchyPowerSeries_integral, ProbabilityTheory.isGaussian_iff_charFun_eq, algHom_ext_iff, has_antideriv_at_fourier_neg, HurwitzZeta.hasSum_expZeta_of_one_lt_re, liftAux_apply_I, CStarAlgebra.linear_combination_nonneg, differentiableAt_complex_iff_differentiableAt_real, two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, CStarAlgebra.norm_smul_two_inv_smul_add_four_unitary, hasDerivAt_fourier_neg, SchwartzMap.fourierInv_lineDerivOp_eq, normSq_ofReal_add_I_mul_sqrt_one_sub, exp_ofReal_mul_I_im, GaussianInt.toComplex_def, ProbabilityTheory.IsGaussian.charFunDual_eq, tsum_exp_neg_quadratic, isAddQuotientCoveringMap_exp, skewAdjoint.negISMul_apply_coe, cot_pi_eq_exp_ratio, I_pow_four, mul_I_re, I_re, HurwitzZeta.hasSum_int_sinKernel, imaginaryPart_eq_neg_I_smul_skewAdjointPart, HurwitzZeta.LSeriesHasSum_exp, bernoulliFourierCoeff_eq, I_im, norm_mul_cos_add_sin_mul_I, isPrimitiveRoot_exp_of_coprime, ModularForm.logDeriv_eta_eq_E2, TemperedDistribution.fourier_lineDerivOp_eq, equivRealProdLm_symm_apply, MeasureTheory.charFunDual_map_const_add, MeasureTheory.charFunDual_map_add_const, realPart_I_smul, Real.fourier_iteratedDeriv, exp_pi_div_two_mul_I, VectorFourier.fourierSMulRight_apply, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, orthonormalBasisOneI_repr_symm_apply, hasDerivAt_circleMap, imaginaryPart_apply_coe, norm_mul_exp_arg_mul_I, NumberField.mixedEmbedding.det_matrixToStdBasis, isPrimitiveRoot_exp_rat_of_even_num, hasDerivAt_fourier, isPrimitiveRoot_exp, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, isIntegral_exp_neg_rat_mul_pi_mul_I, circleMap_zero, Real.fourierIntegral_eq', hasFPowerSeriesOn_cauchy_integral, circleIntegral_div_sub_of_differentiable_on_off_countable, contDiffOn_tsum_cexp, deriv_circleMap, HasStrictDerivAt.complexToReal_fderiv', ProbabilityTheory.HasGaussianLaw.charFun_map_eq, EisensteinSeries.qExpansion_identity_pnat, sqrt_neg_one, Real.iteratedDeriv_fourier, selfAdjoint.star_coe_unitarySelfAddISMul, CliffordAlgebraComplex.toComplex_ι, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', hasSum_cos', jacobiTheta₂'_add_left', jacobiTheta₂'_functional_equation, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, hasSum_arctan_aux, norm_add_mul_I, cos_add_mul_I, DirichletCharacter.norm_LFunction_product_ge_one, measurableEquivPi_symm_apply, I_pow_three, Orientation.kahler_rightAngleRotation_left, tanh_mul_I, enorm_exp_I_mul_ofReal, Real.hasDerivAt_fourierChar, UpperHalfPlane.coe_I, Real.enorm_exp_I_mul_ofReal_sub_one_le, EisensteinSeries.q_expansion_bernoulli, HurwitzZeta.cosKernel_def, I_mul, InnerProductSpace.laplacian_eq_iteratedFDeriv_complexPlane, I_mul_re, wedgeIntegral_add_wedgeIntegral_eq, sqrt_neg_I, integral_boundary_rect_eq_zero_of_differentiable_on_off_countable, sqrt_eq_real_add_ite, fourierCoeffOn_of_hasDerivAt_Ioo, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, SchwartzMap.lineDerivOp_fourierInv_eq, ZMod.completedLFunction_one_sub_odd, sin_eq, Real.fourierChar_apply, lift_symm_apply_coe, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, normSq_add_mul_I, ModularFormClass.qExpansion_coeff_eq_circleIntegral, ProbabilityTheory.isGaussian_iff_charFunDual_eq, fourierIntegral_gaussian, im_eq_sub_conj, HurwitzZeta.hurwitzZeta_one_sub, DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, exp_pi_mul_I, isPrimitiveRoot_exp_rat_of_odd_num, I_mul_I, div_I, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun, exp_sub_pi_mul_I, SchwartzMap.fderivCLM_fourier_eq, nnnorm_exp_ofReal_mul_I, I_sq, norm_I, cos_eq, exp_rat_mul_pi_mul_I_pow_two_mul_den, VectorFourier.fourierPowSMulRight_apply, skewAdjointPart_eq_I_smul_imaginaryPart, MeasureTheory.charFunDual_dirac, EisensteinSeries.G2_eq_tsum_cexp, sin_mul_I, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, sin_add_mul_I, EisensteinSeries.qExpansion_identity, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar_val, HurwitzZeta.hasSum_int_sinZeta, ZMod.toCircle_intCast, ZMod.toCircle_natCast, inner_map_polarization, Real.deriv_fourierChar, exp_eq_exp_re_mul_sin_add_cos, jacobiTheta_eq_tsum_nat, RCLike.norm_to_complex, Orientation.kahler_apply_apply, I_mul_im, bernoulliFourierCoeff_recurrence, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, VectorFourier.fourierIntegral_probChar, IsSelfAdjoint.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary, RCLike.to_complex_nonneg_iff, MeasureTheory.charFun_dirac, HurwitzZeta.sinKernel_def, cot_eq_exp_ratio, fourier_coe_apply', TemperedDistribution.lineDerivOp_fourierInv_eq, ProbabilityTheory.IsGaussian.charFun_eq, polarCoord_symm_apply, Real.hasFDerivAt_fourierChar_neg_bilinear_right, summable_bernoulli_fourier, ZMod.toCircle_apply, circleIntegral.integral_sub_center_inv, SchwartzMap.fourier_fderivCLM_eq, Real.iteratedDeriv_fourierIntegral, adjoin_I, isIntegral_rat_I, hasSum_sin', nnnorm_I, log_neg_I, jacobiTheta₂_add_left', hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, re_add_im, sin_eq_tsum', norm_exp_ofReal_mul_I, Circle.coe_exp, exp_two_pi_mul_I, tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero, integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn, tan_mul_I, Real.hasDerivAt_fourierIntegral, conj_neg_I, rightAngleRotation, fourierCoeffOn_of_hasDerivAt, image_exp_Ioc_eq_sphere, two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, exp_eq_one_iff, Orientation.kahler_rightAngleRotation_right, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, exp_mul_I_antiperiodic, exp_mul_I, differentiableWithinAt_complex_iff_differentiableWithinAt_real, HurwitzZeta.hasSum_int_completedCosZeta, TorusIntegrable.function_integrable, DirichletCharacter.LFunction_isBigO_horizontal, isometryOfOrthonormal_symm_apply, arg_mul_cos_add_sin_mul_I_sub, HurwitzZeta.sinZeta_eq, ProbabilityTheory.IsGaussian.charFunDual_eq', enorm_exp_ofReal_mul_I, Real.probChar_apply, ModularForm.logDeriv_qParam, skewAdjoint.I_smul_neg_I, coe_basisOneI, ProbabilityTheory.charFun_gaussianReal, HurwitzZeta.hasSum_int_completedSinZeta, Real.fourierInv_eq', sinh_mul_I, Real.fourierIntegral_deriv, norm_exp_I_mul_ofReal, exp_mul_I_periodic, range_exp_mul_I, MeasureTheory.charFun_map_const_add, summable_pow_mul_cexp, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq, summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp, isIntegral_int_I, Real.fderiv_fourierChar_neg_bilinear_left_apply, Real.fourier_eq', cosh_mul_I, mul_I_im, conj_I, fourierCoeffOn_of_hasDeriv_right, arg_I, normSq_ofReal_sub_I_mul_sqrt_one_sub, exp_int_mul_two_pi_mul_I, arg_cos_add_sin_mul_I_eq_toIocMod, log_exp_exists, horizontalSegment_eq, Orientation.rightAngleRotation_map_complex, Real.deriv_fourier, CStarAlgebra.exists_sum_four_nonneg, fourierIntegral_gaussian_pi', RCLike.I_to_complex, AnalyticOnNhd.eq_re_add_const_mul_I_of_re_eq_const, selfAdjoint.expUnitary_coe, MeasureTheory.charFun_map_add_const, cpow_ofReal, two_sin, differentiableAt_iteratedDerivWithin_cexp, pi_mul_cot_pi_q_exp, BoundedContinuousFunction.probCharDual_apply, angle_exp_exp, two_cos, restrictScalars_toSpanSingleton', lim_eq_lim_im_add_lim_re, rectangle_eq_convexHull, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
Rectangle 📖	CompOp	2 math math: Convex.rectangle_subset, rectangle_eq_convexHull
addCommGroup 📖	CompOp	290 math math: WeakFEPair.differentiableAt_Λ, NumberField.mixedEmbedding.fundamentalDomain_integerLattice, HurwitzZeta.differentiableAt_hurwitzZetaEven_sub_one_div, ZMod.differentiableAt_LFunction, differentiableAt_sin, differentiable_const_cpow_of_neZero, restrictScalars_one_smulRight', coe_realPart, HurwitzZeta.differentiableAt_hurwitzZeta_sub_one_div, ModularFormClass.differentiableAt_comp_ofComplex, HurwitzZeta.differentiableAt_hurwitzZetaEven, NumberField.canonicalEmbedding.mem_span_latticeBasis, PeriodPair.hasSumLocallyUniformly_weierstrassP, PeriodPair.deriv_weierstrassP, PeriodPair.coeff_weierstrassPExceptSeries, PeriodPair.derivWeierstrassPExcept_sub, realPart_ofReal, PeriodPair.ω₁_div_two_notMem_lattice, circleIntegral_def_Icc, ZMod.LFunction_one_sub, circleIntegrable_iff, DirichletCharacter.continuousOn_neg_logDeriv_LFunction_of_nontriv, Real.fderiv_fourierChar_neg_bilinear_right_apply, ZMod.dft_even_iff, Unitization.real_cfcₙ_eq_cfc_inr, HasFDerivWithinAt.complexOfReal, reLm_coe, ProbabilityTheory.hasDerivAt_integral_pow_mul_exp, NumberField.mixedEmbedding.mem_idealLattice, re_balance, hasStrictDerivAt_tan, PeriodPair.eqOn_deriv_weierstrassPExcept_derivWeierstrassPExcept, PeriodPair.weierstrassP_coe, DirichletCharacter.LSeries_twist_vonMangoldt_eq, deriv_cos, instFiniteDimensionalReal, not_differentiableAt_Gamma_neg_nat, differentiableOn_dslope, differentiableOn_iteratedDerivWithin_cotTerm, PeriodPair.hasSum_derivWeierstrassPExcept, differentiableAt_GammaAux, HarmonicAt.analyticAt_complex_partial, DirichletCharacter.deriv_LFunction_eq_deriv_LSeries, StrongFEPair.differentiable_Λ, ZMod.differentiableAt_completedLFunction, hasDerivAt_GammaIntegral, deriv_ofReal_cpow_const, analyticOnNhd_univ_iff_differentiable, PeriodPair.latticeBasis_zero, TemperedDistribution.instLineDerivLeftSMulReal, hasStrictFDerivAt_cpow', PeriodPair.ω₂_div_two_notMem_lattice, HurwitzZeta.differentiable_expZeta_of_ne_zero, PeriodPair.hasSum_derivWeierstrassP, PeriodPair.compl_lattice_diff_singleton_mem_nhds, HurwitzZeta.differentiable_hurwitzZetaOdd, LSeries_deriv_eqOn, HurwitzZeta.differentiable_hurwitzZeta_sub_hurwitzZeta, PeriodPair.hasSumLocallyUniformly_derivWeierstrassP, deriv_Gamma_add_one, PeriodPair.mem_lattice, PeriodPair.analyticOnNhd_derivWeierstrassPExcept, DirichletCharacter.differentiableAt_LFunction, hasDerivAt_Gammaℝ_one, hasFDerivAt_jacobiTheta₂_term, LSeries_deriv, NumberField.mixedEmbedding.fundamentalDomain_idealLattice, differentiable_exp, re_comp_balance, summable_jacobiTheta₂_term_fderiv_iff, NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis, PeriodPair.isClosed_lattice, NumberField.mixedEmbedding.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIdealLattice, hasStrictFDerivAt_cpow, differentiable_iteratedDeriv_sin, hasStrictDerivAt_sinh, ofReal_balance, PeriodPair.periodic_derivWeierstrassP, HarmonicAt.differentiableAt_complex_partial, EisensteinSeries.div_linear_zpow_differentiableOn, ZMod.differentiable_completedLFunction, NumberField.mixedEmbedding.span_idealLatticeBasis, deriv_Gamma_nat, IsExactOn.differentiableOn, hasDerivAt_jacobiTheta₂_fst, ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div, PeriodPair.differentiableOn_derivWeierstrassP, hasDerivAt_sinh, PeriodPair.weierstrassPExcept_def, hasSum_jacobiTheta₂_term_fderiv, NumberField.mixedEmbedding.mem_span_latticeBasis, hasDerivAt_Gammaℂ_one, det_conjAe, hasStrictDerivAt_const_cpow, integrableOn_Ioi_deriv_ofReal_cpow, hasFDerivAt_cpow, differentiableAt_exp, differentiable_Gammaℝ_inv, differentiableAt_jacobiTheta₂_fst, iter_deriv_exp, differentiableAt_tan, ModularForm.differentiableAt_eta_tprod, has_antideriv_at_fourier_neg, selfAdjointEquiv_apply, PeriodPair.hasSumLocallyUniformly_weierstrassPExcept, equivRealProdLm_apply, hasDerivAt_cos, differentiableAt_complex_iff_differentiableAt_real, cfcHom_real_eq_restrict, Polynomial.eq_centerMass_of_eval_derivative_eq_zero, hasDerivAt_fourier_neg, hasStrictDerivAt_log, NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm, ProbabilityTheory.hasDerivAt_complexMGF, PeriodPair.derivWeierstrassP_sub_coe, HurwitzZeta.differentiableAt_completedHurwitzZetaEven, rank_real_complex', HurwitzZeta.differentiableAt_one_completedHurwitzZetaEven_sub_completedHurwitzZetaEven, TemperedDistribution.instLineDerivSMulReal, analyticOn_iff_differentiableOn, ofReal_comp_balance, differentiable_sin, PeriodPair.weierstrassP_add_coe, equivRealProdLm_symm_apply, equivRealProdLm_symm_apply_im, PeriodPair.instProperSpaceSubtypeComplexMemSubmoduleIntLattice, differentiableOn_compl_singleton_and_continuousAt_iff, hasDerivAt_logTaylor, DirichletCharacter.differentiable_LFunction, NumberField.mixedEmbedding.span_latticeBasis, hasDerivAt_circleMap, im_balance, ModularForm.differentiableAt_eta_of_mem_upperHalfPlaneSet, hasDerivAt_fourier, ModularFormClass.differentiableAt_cuspFunction, deriv_circleMap, PeriodPair.latticeBasis_one, hasStrictDerivAt_cos, PeriodPair.weierstrassP_sub_coe, analyticOn_univ_iff_differentiable, ArithmeticFunction.LSeries_vonMangoldt_eq, DirichletCharacter.IsPrimitive.fourierTransform_eq_inv_mul_gaussSum, DirichletCharacter.differentiableAt_completedLFunction, NumberField.mixedEmbedding.euclidean.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIntegerLattice, linearEquiv_det_conjAe, PeriodPair.hasSum_weierstrassP, HasFDerivAt.complexOfReal_hasFDerivAt, PeriodPair.coeff_weierstrassPSeries, PeriodPair.isOpen_compl_lattice_diff, deriv_cosh, PeriodPair.derivWeierstrassP_coe, hasStrictDerivAt_sin, HasDerivAt.ofReal_comp, DifferentiableAt.ofReal_cpow_const, hasFDerivAt_jacobiTheta₂, im_comp_balance, deriv_tan, hasDerivAt_cosh, Unitization.nnreal_cfcₙ_eq_cfc_inr, HurwitzZeta.differentiableAt_hurwitzZeta, DirichletCharacter.deriv_LFunctionTrivChar₁_apply_of_ne_one, LSeries_differentiableOn, differentiableAt_riemannZeta, cfcₙHom_real_eq_restrict, PeriodPair.latticeEquiv_symm_apply, equivRealProdLm_symm_apply_re, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, hasDerivAt_sin, Real.hasDerivAt_fourierChar, PeriodPair.deriv_derivWeierstrassPExcept_self, hasStrictDerivAt_exp, hasStrictDerivAt_cosh, isConservativeOn_and_continuousOn_iff_isDifferentiableOn, differentiableAt_const_cpow_of_neZero, coe_selfAdjointEquiv, differentiable_one_div_Gamma, imaginaryPart_ofReal, ZMod.completedLFunction_one_sub_odd, not_differentiableAt_Gamma_zero, NumberField.mixedEmbedding.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIntegerLattice, ProbabilityTheory.differentiableOn_complexMGF, PeriodPair.deriv_weierstrassPExcept_same, PeriodPair.analyticOnNhd_derivWeierstrassP, EulerSine.antideriv_cos_comp_const_mul, HurwitzZeta.differentiableAt_completedCosZeta, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, EulerSine.antideriv_sin_comp_const_mul, differentiableAt_cosh, differentiableAt_Gamma_nat_add_one, deriv_cos', PeriodPair.instDiscreteTopologySubtypeComplexMemSubmoduleIntLattice, qExpansion_neg, isBigO_deriv_ofReal_cpow_const_atTop, hasDerivAt_ofReal_cpow_const, selfAdjointEquiv_symm_apply, differentiableAt_jacobiTheta₂_snd, finrank_real_complex, hasStrictDerivAt_cpow_const, mellin_differentiableAt_of_isBigO_rpow_exp, finrank_real_complex_fact, HurwitzZeta.differentiable_cosZeta_of_ne_zero, deriv_cpow_const, differentiableAt_Gamma_one, Real.differentiable_fourierChar, hasDerivAt_Gamma_one, PeriodPair.hasSum_weierstrassPExcept, ProbabilityTheory.hasDerivAt_iteratedDeriv_complexMGF, differentiable_completedZeta₀, differentiable_sinh, Real.deriv_fourierChar, differentiable_iteratedDeriv_sinh, differentiableAt_sinh, hasDerivAt_ofReal_cpow_const', hasDerivAt_Gamma_one_half, Function.Periodic.differentiable_qParam, PeriodPair.differentiableOn_weierstrassPExcept, analyticOnNhd_iff_differentiableOn, Real.differentiable_fourierChar_neg_bilinear_right, differentiableAt_log, differentiableAt_completedZeta, PeriodPair.derivWeierstrassPExcept_add_coe, PeriodPair.differentiableOn_weierstrassP, EisensteinSeries.eisSummand_extension_differentiableOn, deriv_sinh, deriv_sin, Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv, PeriodPair.hasSum_sumInvPow, hasDerivAt_tan, conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj, ModularFormClass.differentiableOn_cuspFunction_ball, ZMod.differentiable_completedLFunction₀, PeriodPair.periodic_weierstrassP, differentiableAt_Gamma, HurwitzZeta.differentiableAt_cosZeta, NumberField.mixedEmbedding.euclidean.finrank, Real.differentiable_fourierChar_neg_bilinear_left, mellin_differentiableAt_of_isBigO_rpow, PeriodPair.mul_ω₁_add_mul_ω₂_mem_lattice, differentiable_cosh, ZMod.completedLFunction_one_sub_even, differentiable_iteratedDeriv_cos, isTheta_deriv_ofReal_cpow_const_atTop, differentiable_cos, CircleIntegrable.out, LSeries_hasDerivAt, UpperHalfPlane.mdifferentiableAt_iff, PeriodPair.analyticOnNhd_weierstrassPExcept, deriv_circleMap_eq_zero_iff, hasDerivAt_exp, DirichletCharacter.fourierTransform_eq_gaussSum_mulShift, hasDerivAt_log_sub_logTaylor, ZMod.dft_odd_iff, hasDerivAt_log, ZMod.differentiable_LFunction_of_sum_zero, PeriodPair.lattice_eq_span_range_basis, PeriodPair.ω₁_mem_lattice, ZMod.LFunction_dft, PeriodPair.analyticOnNhd_weierstrassP, differentiableAt_cos, HurwitzZeta.differentiable_completedHurwitzZetaEven₀, HurwitzZeta.differentiable_completedHurwitzZetaOdd, deriv_exp, Real.fderiv_fourierChar_neg_bilinear_left_apply, PeriodPair.ω₂_mem_lattice, DirichletCharacter.differentiable_LFunctionTrivChar₁, UpperHalfPlane.mdifferentiable_iff, analyticAt_iff_eventually_differentiableAt, HurwitzZeta.differentiable_completedSinZeta, HurwitzZeta.differentiable_hurwitzZetaEven_sub_hurwitzZetaEven, imLm_coe, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, PeriodPair.derivWeierstrassPExcept_def, LSeries.hasDerivAt_term, WeakFEPair.differentiable_Λ₀, DirichletCharacter.continuousOn_neg_logDeriv_LFunctionTrivChar₁, DirichletCharacter.differentiable_completedLFunction, HurwitzZeta.differentiableAt_expZeta, PeriodPair.finrank_lattice, PeriodPair.derivWeierstrassP_add_coe, rank_real_complex, HurwitzZeta.differentiableAt_sinZeta, differentiableAt_jacobiTheta, differentiableAt_iteratedDerivWithin_cexp, PeriodPair.weierstrassPExcept_add, hasDerivAt_Gamma_nat, differentiable_circleMap, HurwitzZeta.differentiable_completedCosZeta₀, restrictScalars_toSpanSingleton', differentiable_iteratedDeriv_cosh, PeriodPair.differentiableOn_derivWeierstrassPExcept
addGroupWithOne 📖	CompOp	17 math math: exp_nsmul, selfAdjointEquiv_apply, ofReal_nsmul, expMonoidHom_apply, coe_imAddGroupHom, instIsAddTorsionFree, im_nsmul, coe_selfAdjointEquiv, ofReal_zsmul, selfAdjointEquiv_symm_apply, ofReal_comp_nsmul, ofReal_comp_zsmul, re_zsmul, re_nsmul, im_zsmul, coe_reAddGroupHom, instCharZero
commRing 📖	CompOp	283 math math: ArithmeticFunction.vonMangoldt.residueClass_eq, NumberField.basisMatrix_eq_embeddingsMatrixReindex, TemperedDistribution.lineDerivOpCLM_eq, Polynomial.Chebyshev.S_two_mul_complex_cos, MeasureTheory.Lp.toTemperedDistributionCLM_apply, restrictScalars_one_smulRight', coe_realPart, SlashInvariantForm.coe_prodEqualWeights, SchwartzMap.integral_sesq_fourier_fourier, Real.fderiv_fourierIntegral, CStarModule.innerₛₗ_apply, DirichletCharacter.LSeries_eulerProduct_tprod, Polynomial.Chebyshev.complex_ofReal_eval_S, Polynomial.mahlerMeasure_eq_leadingCoeff_mul_prod_roots, realPart_ofReal, HasDerivWithinAt.complexToReal_fderiv', Real.fourier_continuousLinearMap_apply, Unitization.real_cfcₙ_eq_cfc_inr, SchwartzMap.lineDerivOp_fourier_eq, star_def, SchwartzMap.integral_bilin_fourierInv_eq, AddChar.doubleDualEmb_injective, TemperedDistribution.lineDerivOp_fourier_eq, HasProdUniformlyOn_sineTerm_prod_on_compact, reLm_coe, Polynomial.one_le_prod_max_one_norm_roots, UnitAddTorus.mFourierSubalgebra_coe, TemperedDistribution.smulLeftCLM_compL_smulLeftCLM, im_expect, Polynomial.Chebyshev.complex_ofReal_eval_T, ModularForm.prod_slash, isQuotientCoveringMap_npow, CStarModule.innerSL_apply, UnitAddTorus.mFourierSubalgebra_closure_eq_top, UpperHalfPlane.prod_eq_zero_iff, toMatrix_rotation, ModularForm.prod_slash_sum_weights, gelfandStarTransform_apply_apply, CStarMatrix.toCLM_injective, DirichletCharacter.LFunction_changeLevel, gelfandTransform_bijective, SchwartzMap.integral_bilin_fourierIntegral_eq, Real.fourierIntegral_continuousMultilinearMap_apply, isPrimitiveRoot_exp_rat, VectorFourier.fourierPowSMulRight_eq_comp, mem_rootsOfUnity, TemperedDistribution.smulLeftCLM_neg, cexp_tsum_eq_tprod, SchwartzMap.integral_sesq_fourier_eq, Real.fderiv_fourier, MeasureTheory.charFunDual_pi', CStarMatrix.norm_def', VectorFourier.fourierIntegral_continuousLinearMap_apply, SchwartzMap.fourier_lineDerivOp_eq, Polynomial.Chebyshev.complex_ofReal_eval_C, CStarMatrix.inner_toCLM_conjTranspose_left, TemperedDistribution.fourierInv_lineDerivOp_eq, Real.fourier_iteratedFDeriv, CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM, TemperedDistribution.instFourierSMul, VectorFourier.hasFDerivAt_fourierIntegral, VectorFourier.fourierIntegral_iteratedFDeriv, BoundedContinuousFunction.mem_charPoly, spectrum.gelfandTransform_eq, ArithmeticFunction.vonMangoldt.residueClass_apply, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, NumberField.Units.map_complexEmbedding_torsion, HasSum.cexp, Real.fourierIntegral_continuousLinearMap_apply, ModularForm.coe_prodEqualWeights, gelfandStarTransform_symm_apply, DirichletCharacter.LSeries_changeLevel, Real.hasFDerivAt_fourier, Polynomial.int_cyclotomic_rw, VectorFourier.iteratedFDeriv_fourierIntegral, tendsto_logDeriv_euler_sin_div, EulerSine.sin_pi_mul_eq, VonNeumannAlgebra.centralizer_centralizer', StarAlgebra.elemental.bijective_characterSpaceToSpectrum, isPrimitiveRoot_iff, gelfandTransform_map_star, multipliable_of_summable_log, norm_exp_sub_sum_le_norm_mul_exp, liftAux_neg_I, isPrimitiveRoot_exp_of_isCoprime, Polynomial.Chebyshev.T_complex_cosh, Polynomial.Chebyshev.C_two_mul_complex_cosh, MeasureTheory.charFun_pi, SchwartzMap.convolution_apply, Polynomial.prod_max_one_norm_roots_le_mahlerMeasure_of_one_le_leadingCoeff, ofReal_prod, SchwartzMap.integral_bilin_fourier_eq, SchwartzMap.fourierInv_apply_eq, HasDerivAt.complexToReal_fderiv', SchwartzMap.toTemperedDistributionCLM_apply_apply, fourierSubalgebra_separatesPoints, TemperedDistribution.smulLeftCLM_sub, DirichletCharacter.LFunctionTrivChar_eq_mul_riemannZeta, ZMod.isPrimitive_stdAddChar, HasProdLocallyUniformlyOn_euler_sin_prod, Polynomial.rootSet_derivative_subset_convexHull_rootSet, VonNeumannAlgebra.mem_carrier, ModularForm.differentiableAt_eta_tprod, selfAdjointEquiv_apply, Polynomial.one_le_mahlerMeasure_of_ne_zero, StarSubalgebra.spectrum_eq, equivRealProdLm_apply, CStarMatrix.toCLM_apply_single, Real.fourier_fderiv, Polynomial.logMahlerMeasure_eq_log_leadingCoeff_add_sum_log_roots, Polynomial.eq_centerMass_of_eval_derivative_eq_zero, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, VectorFourier.fderiv_fourierIntegral, SchwartzMap.fourierInv_lineDerivOp_eq, VectorFourier.integral_fourierIntegral_swap, tendsto_euler_sin_prod, CStarMatrix.toCLM_apply_eq_sum, AddChar.doubleDualEmb_bijective, rank_real_complex', Polynomial.prod_mahlerMeasure_eq_mahlerMeasure_prod, ofReal_sum, SchwartzMap.fourier_evalCLM_eq, SlashInvariantForm.coe_prod, CStarMatrix.toCLM_apply, isPrimitiveRoot_exp_of_coprime, TemperedDistribution.fourier_lineDerivOp_eq, equivRealProdLm_symm_apply, equivRealProdLm_symm_apply_im, coe_basisOneI_repr, PositiveLinearMap.gnsStarAlgHom_apply, surjective_rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar, TemperedDistribution.lineDerivOp_toTemperedDistributionCLM_eq, NumberField.discr_eq_basisMatrix_det_sq, NumberField.mixedEmbedding.det_matrixToStdBasis, ofReal_expect, isPrimitiveRoot_exp_rat_of_even_num, AddChar.starComp_apply, Real.fourier_bilin_convolution_eq, CStarMatrix.mul_entry_mul_eq_inner_toCLM, isPrimitiveRoot_exp, ProbabilityTheory.iIndepFun_iff_charFunDual_pi, SchwartzMap.integral_sesq_fourierIntegral_eq, VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral', qExpansion_add, HasStrictDerivAt.complexToReal_fderiv', DirichletCharacter.IsPrimitive.fourierTransform_eq_inv_mul_gaussSum, instStarModuleReal, TemperedDistribution.smulLeftCLM_add, Polynomial.Chebyshev.U_complex_cosh, AddChar.coe_doubleDualEquiv, fourierSubalgebra_coe, multipliableUniformlyOn_of_clog, Unitization.nnreal_cfcₙ_eq_cfc_inr, NumberField.inverse_basisMatrix_mulVec_eq_repr, Polynomial.Chebyshev.T_complex_cos, riemannZeta_eulerProduct, Real.fourierIntegral_fderiv, equivRealProdLm_symm_apply_re, UnitAddTorus.mFourierSubalgebra_separatesPoints, AddChar.starComp_eq_inv, exp_bound, Polynomial.sum_derivRootWeight_pos, multipliable_sineTerm, MeasureTheory.charFunDual_eq_pi_iff, DirichletCharacter.LSeries_eulerProduct_hasProd, card_primitiveRoots, Real.iteratedFDeriv_fourier, im_sum, Real.hasFDerivAt_fourierIntegral, coe_selfAdjointEquiv, Polynomial.Chebyshev.S_two_mul_complex_cosh, imaginaryPart_ofReal, SchwartzMap.lineDerivOp_fourierInv_eq, StarSubalgebra.mem_spectrum_iff, MulChar.star_eq_inv, ProbabilityTheory.iIndepFun_iff_charFunDual_pi', one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, ModularForm.coe_prod, Algebra.trace_complex_apply, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, TemperedDistribution.instFourierInvSMul, isPrimitiveRoot_exp_rat_of_odd_num, selfAdjointEquiv_symm_apply, SchwartzMap.fderivCLM_fourier_eq, ProbabilityTheory.iIndepFun_iff_charFun_pi, finrank_real_complex, Polynomial.Chebyshev.complex_ofReal_eval_U, VectorFourier.integral_bilin_fourierIntegral_eq_flip, finrank_real_complex_fact, TemperedDistribution.instLineDerivSMulComplex, euler_sineTerm_tprod, exp_bound', Polynomial.Chebyshev.U_complex_cos, MeasureTheory.charFunDual_eq_pi_iff', ModularForm.coe_norm, Real.fourierIntegral_continuousLinearMap_apply', VonNeumannAlgebra.coe_toStarSubalgebra, logDeriv_prod_sineTerm_eq_sum_cotTerm, Polynomial.Chebyshev.C_two_mul_complex_cos, StarSubalgebra.coe_isUnit, GammaSeq_mul, fourierSubalgebra_closure_eq_top, ModularForm.prod_fintype_slash, GaussianFourier.integral_cexp_neg_sum_mul_add, Unitization.complex_cfcₙ_eq_cfc_inr, VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral, rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar_val, AddChar.doubleDualEquiv_symm_doubleDualEmb_apply, toMatrix_conjAe, TemperedDistribution.derivCLM_toTemperedDistributionCLM_eq, Real.iteratedFDeriv_fourierIntegral, VectorFourier.fourierIntegral_continuousMultilinearMap_apply, MeasureTheory.Lp.ker_toTemperedDistributionCLM_eq_bot, MeasureTheory.Lp.toTemperedDistribution_toLp_eq, rootsOfUnityCircleEquiv_apply, AddChar.FiniteField.primitiveChar_to_Complex_isPrimitive, card_rootsOfUnity, TemperedDistribution.smulLeftCLM_smulLeftCLM_apply, hasProd_of_hasSum_log, rank_rat_complex, Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv, TemperedDistribution.lineDerivOp_fourierInv_eq, re_expect, DirichletCharacter.LFunctionTrivChar_residue_one, exp_sum, Real.fourier_continuousMultilinearMap_apply, AddChar.doubleDualEmb_inj, norm_exp_sub_sum_le_exp_norm_sub_sum, TemperedDistribution.smulLeftCLM_const, MeasureTheory.charFunDual_pi, SchwartzMap.fourier_fderivCLM_eq, SchwartzMap.fourier_convolution_apply, TemperedDistribution.smulLeftCLM_sum, norm_prod, gelfandTransform_isometry, AddChar.doubleDualEmb_eq_zero, betaIntegral_eval_nat_add_one_right, VectorFourier.fourierPowSMulRight_iteratedFDeriv_fourierIntegral, riemannZeta_eulerProduct_tprod, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, AddChar.doubleDualEmb_doubleDualEquiv_symm_apply, TemperedDistribution.smulLeftCLM_smul, Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero, hasProdUniformlyOn_of_clog, TemperedDistribution.derivCLM_apply_apply, TorusIntegrable.function_integrable, DirichletCharacter.fourierTransform_eq_gaussSum_mulShift, Polynomial.finite_mahlerMeasure_le, tendsto_euler_sin_prod', coe_basisOneI, multipliableUniformlyOn_euler_sin_prod_on_compact, riemannZeta_eulerProduct_hasProd, isCauSeq_exp, VectorFourier.hasFDerivAt_fourierChar_smul, BoundedContinuousFunction.separatesPoints_charPoly, DirichletCharacter.LSeries_eulerProduct, re_sum, AddChar.zmodAddEquiv_apply, MeasureTheory.charFun_eq_pi_iff, exp_multiset_sum, VectorFourier.fourierIntegral_fderiv, VectorFourier.norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le, Real.fourierIntegral_continuousMultilinearMap_apply', Real.fourierIntegral_iteratedFDeriv, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, imLm_coe, SlashInvariantForm.coe_norm, TemperedDistribution.laplacianCLM_apply, Polynomial.card_mahlerMeasure_le_prod, MeasureTheory.Lp.toTemperedDistribution_smul_eq, nonempty_linearEquiv_real, ofReal_choose, rank_real_complex, CStarMatrix.norm_def, ModularForm.multipliableLocallyUniformlyOn_eta, TemperedDistribution.smulLeftCLM_apply_apply, BoundedContinuousFunction.char_mem_charPoly, multipliable_one_add_of_summable, CStarMatrix.inner_toCLM_conjTranspose_right, CStarMatrix.toCLM_apply_single_apply, Unitization.cfcₙ_eq_cfc_inr, restrictScalars_toSpanSingleton', Polynomial.int_cyclotomic_spec
equivRealProd 📖	CompOp	10 math math: lipschitz_equivRealProd, equivRealProd_apply, equivRealProd_symm_apply, equivRealProd_apply_le', antilipschitz_equivRealProd, preimage_equivRealProd_prod, NumberField.mixedEmbedding.mixedSpaceToRealMixedSpace_apply, polarCoord_apply, isUniformEmbedding_equivRealProd, equivRealProd_apply_le
equivRealProdAddHom 📖	CompOp	4 math math: equivRealProdAddHom_symm_apply, equivRealProdAddHom_symm_apply_re, equivRealProdAddHom_symm_apply_im, equivRealProdAddHom_apply
im 📖	CompOp	260 math math: im_qsmul, TendstoUniformly.im, mul_im, mellin_eq_fourier, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, isOpenMap_im, NumberField.mixedEmbedding.convexBodyLT'_mem, areaForm, GaussianInt.intCast_im, norm_le_abs_re_add_abs_im, CuspFormClass.zero_at_infty_comp_ofComplex, AnalyticOn.im_ofReal, add_im, exp_re, uniformlyContinuous_im, div_ofNat_im, arg_of_re_nonneg, nndist_self_conj, mul_re, equivRealProd_apply, UpperHalfPlane.specialLinearGroup_apply, closure_preimage_im, Quaternion.imI_coeComplex, UpperHalfPlane.canLift, IsExpCmpFilter.isBigO_im_pow_re, im_expect, UpperHalfPlane.moebius_im, Function.Periodic.norm_qParam_lt_iff, toMatrix_rotation, neg_pi_div_two_lt_arg_iff, arg_le_pi_div_two_iff, im_nnqsmul, measurableEquivPi_apply, uniformContinuous_im, cosh_ofReal_im, UpperHalfPlane.glPos_smul_def, sub_im, AnalyticAt.harmonicAt_im, log_exp_eq_re_add_toIocMod, tan_arg, NumberField.mixedEmbedding.stdBasis_apply_isComplex_snd, liftAux_apply, ratCast_im, natCast_im, measurableEquivRealProd_apply, frontier_setOf_im_lt, ofReal_mul', riemannZeta_im_eq_zero_of_one_lt, PhragmenLindelof.eqOn_quadrant_II, sq_norm_sub_sq_re, abs_re_eq_norm, norm_cpow_of_ne_zero, PhragmenLindelof.eq_zero_on_quadrant_IV, im_le_norm, closure_setOf_lt_im, dist_self_conj, div_ofNat, orthonormalBasisOneI_repr_apply, norm_cpow_of_imp, summable_jacobiTheta₂_term_fderiv_iff, convex_halfSpace_im_gt, norm_jacobiTheta₂_term, PhragmenLindelof.eqOn_quadrant_I, abs_im_le_norm, div_im, im_nnratCast, PhragmenLindelof.eqOn_quadrant_III, tan_ofReal_im, EisensteinSeries.div_linear_zpow_differentiableOn, sq_norm_sub_sq_im, arg_neg_eq_arg_add_pi_iff, abs_im_eq_norm, interior_preimage_im, log_exp_eq_sub_toIocDiv, eta, Function.Periodic.im_invQParam, GaussianInt.toComplex_im_div, sub_conj, conj_im, arg_lt_pi_div_two_iff, interior_setOf_le_im, zero_im, sq_nonneg_iff, convex_halfSpace_im_lt, equivRealProdAddHom_symm_apply_im, UpperHalfPlane.im_smul, neg_iff, frontier_preimage_im, equivRealProdLm_apply, div_intCast, nonpos_iff, NumberField.mixedEmbedding.exists_ne_zero_mem_ringOfIntegers_lt', exp_ofReal_mul_I_im, arg_eq_pi_div_two_iff, isBigO_at_im_infty_jacobiTheta_sub_one, frontier_setOf_lt_im, exp_ofReal_im, slitPlane_eq_union, mul_I_re, nndist_of_re_eq, interior_setOf_im_le, ext_iff, abs_im_lt_norm, UpperHalfPlane.ofComplex_apply_eq_ite, neg_pi_lt_log_im, I_im, LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re, equivRealProdLm_symm_apply_im, coe_basisOneI_repr, GaussianInt.to_real_im, measurable_im, im_balance, IsSelfAdjoint.im_eq_zero_of_mem_spectrum, div_re, mellin_eq_fourierIntegral, tanh_ofReal_im, Algebra.leftMulMatrix_complex, AnalyticWithinAt.im_ofReal, isBigO_im_sub_im, convex_halfSpace_im_ge, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', div_ofReal, coe_imAddGroupHom, im_comp_balance, RCLike.complexRingEquiv_symm_apply, abs_im_div_norm_le_one, UpperHalfPlane.IsZeroAtImInfty.zero_at_infty_comp_ofComplex, log_im, equivRealProdAddHom_apply, isOpen_im_gt_EReal, UpperHalfPlane.im_pos_of_dist_center_le, arg_neg_iff, TendstoUniformlyOn.im, Function.Periodic.invQParam_tendsto, im_nsmul, IsExpCmpFilter.isLittleO_im_pow_exp_re, Function.Periodic.im_invQParam_pos_of_norm_lt_one, I_mul, I_mul_re, arg_eq_zero_iff, wedgeIntegral_add_wedgeIntegral_eq, GaussianInt.toComplex_im, GaussianInt.im_toComplex, im_sum, sqrt_eq_real_add_ite, nndist_conj_self, pos_iff, NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt', sin_eq, Function.Periodic.qParam_tendsto, div_ratCast_im, sinh_ofReal_im, PhragmenLindelof.eq_zero_on_quadrant_II, im_mul_ofReal, im_eq_sub_conj, lim_im, div_natCast_im, isQuotientMap_im, UnitDisc.im_coe, isOpen_im_lt_EReal, PhragmenLindelof.eqOn_quadrant_IV, cpow_ofReal_im, cos_eq, isCauSeq_im, im_ofReal_mul, inv_im, isometryOfOrthonormal_apply, frontier_setOf_le_re_and_le_im, Orientation.areaForm_map_complex, Function.Periodic.norm_qParam, ModularFormClass.bounded_at_infty_comp_ofComplex, im_surjective, hasSum_im, RCLike.im_to_complex, UpperHalfPlane.im_inv_neg_coe_pos, exp_eq_exp_re_mul_sin_add_cos, neg_pi_div_two_le_arg_iff, canLift, div_ofReal_im, IsExpCmpFilter.abs_im_pow_eventuallyLE_exp_re, conj_eq_iff_im, I_mul_im, arg_eq_nhds_of_re_pos, imaginaryPart_smul, sin_ofReal_im, EisensteinSeries.eisSummand_extension_differentiableOn, EisensteinSeries.eisensteinSeries_tendstoLocallyUniformlyOn, ofReal_im, equivRealProdCLM_symm_apply_im, RCLike.im_eq_complex_im, Quaternion.coeComplex_imI, MeasureTheory.SignedMeasure.toComplexMeasure_apply_im, re_add_im, nonneg_iff, im_eq_zero_iff_isSelfAdjoint, arg_nonneg_iff, normSq_apply, convex_halfSpace_im_le, UpperHalfPlane.tendsto_coe_atImInfty, arg_exp, div_intCast_im, frontier_setOf_le_re_and_im_le, im_sq_le_normSq, PhragmenLindelof.eq_zero_on_quadrant_I, realPart_smul, norm_eq_sqrt_sq_add_sq, GaussianFourier.verticalIntegral_norm_le, le_def, div_natCast, edist_of_re_eq, imCLM_apply, norm_cpow_le, log_im_le_pi, intCast_im, frontier_setOf_le_im, im_zsmul, abs_cpow_inv_two_im, UpperHalfPlane.im_pnat_div_pos, UpperHalfPlane.coe_im, norm_le_sqrt_two_mul_max, isHomeomorphicTrivialFiberBundle_im, UpperHalfPlane.tendsto_comap_im_ofComplex, cos_ofReal_im, upperHalfPlane_inter_integerComplement, frontier_setOf_im_le, range_im, mem_reProdIm, dist_of_re_eq, hasSum_iff, smul_im, UpperHalfPlane.coe_im_pos, PhragmenLindelof.eq_zero_on_quadrant_III, summable_jacobiTheta₂'_term_iff, UpperHalfPlane.mdifferentiable_iff, UpperHalfPlane.smulAux'_im, arg_neg_eq_arg_sub_pi_iff, mul_I_im, AnalyticOnNhd.im_ofReal, imLm_coe, arg_eq_pi_iff, neg_im, sin_arg, exp_im, dist_conj_self, div_ratCast, im_tsum, arg_eq_neg_pi_div_two_iff, continuous_im, equivRealProdCLM_apply, summable_jacobiTheta₂_term_iff, GaussianInt.toComplex_div_im, im_ofNat, one_im, norm_mul_sin_arg, dist_eq_re_im, lt_def, AnalyticAt.im_ofReal, rectangle_eq_convexHull, closure_setOf_im_lt, ModularGroup.coe_truncatedFundamentalDomain
imAddGroupHom 📖	CompOp	1 math math: coe_imAddGroupHom
instAdd 📖	CompOp	435 math math: HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip_eps, Real.circleAverage_map_add_const, Polynomial.Chebyshev.S_two_mul_complex_cos, verticalSegment_eq, RCLike.sqrt_eq_ite, hasSum_mellin_pi_mul_sq', norm_log_one_sub_inv_add_logTaylor_neg_le, one_add_tan_sq_mul_cos_sq_eq_one, EisensteinSeries.linear_inv_isBigO_left, GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I, PeriodPair.hasFPowerSeriesAt_weierstrassPExcept, mk_eq_add_mul_I, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, exp_eq_exp_iff_exists_int, re_eq_add_conj, GaussianInt.toComplex_def', integral_boundary_rect_eq_zero_of_differentiableOn, Gamma_nat_eq_factorial, add_im, digamma_apply_add_one, equivRealProdAddHom_symm_apply, NumberField.mixedEmbedding.convexBodySumFun_add_le, ZMod.LFunction_one_sub, one_add_cpow_hasFPowerSeriesAt_zero, exp_add_mul_I, arg_cos_add_sin_mul_I_coe_angle, EisensteinSeries.summand_bound_of_mem_verticalStrip, HurwitzZeta.jacobiTheta₂''_add_left, hasSum_one_div_nat_pow_mul_fourier, betaIntegral_scaled, cos_eq_zero_iff, cos_sub, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, cos_antiperiodic, cos_add_nat_mul_two_pi, HasProdUniformlyOn_sineTerm_prod_on_compact, ProbabilityTheory.hasDerivAt_integral_pow_mul_exp, sin_eq_neg_one_iff, UpperHalfPlane.specialLinearGroup_apply, HurwitzZeta.cosZeta_neg_two_mul_nat_add_one, fourierIntegral_gaussian_innerProductSpace', sqrt_I, fourier_gaussian_pi', HasStrictDerivAt.cpow, GaussianFourier.integral_cexp_neg_mul_sq_norm_add, sum_cauchyPowerSeries_eq_integral, Gamma_add_one, norm_one_add_mul_inv_le, SlashInvariantForm.add_apply, norm_cos_add_sin_mul_I, hasSum_nat_jacobiTheta, DirichletCharacter.norm_LSeries_product_ge_one, exp_add, gelfandStarTransform_naturality, gelfandStarTransform_apply_apply, tendsto_mul_log_one_add_of_tendsto, ModularForm.tsum_logDeriv_eta_q, norm_log_one_add_le, EisensteinSeries.summable_left_one_div_linear_sub_one_div_linear, HasDerivAt.cpow, HurwitzZeta.oddKernel_def', cos_eq_two_mul_tan_half_div_one_sub_tan_half_sq, sinh_add_cosh, one_div_Gamma_eq_self_mul_one_div_Gamma_add_one, ModularGroup.coe_T_zpow_smul_eq, hasDerivAt_GammaIntegral, PeriodPair.ite_eq_one_sub_sq_mul_weierstrassP, exp_antiperiodic, exp_add_pi_mul_I, log_exp_eq_re_add_toIocMod, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace, circleMap_eq_circleMap_iff, equivRealProd_symm_apply, UpperHalfPlane.re_add_im, RCLike.complexRingEquiv_apply, arg_cos_add_sin_mul_I, cos_eq_iff_quadratic, sinh_three_mul, PeriodPair.hasFPowerSeriesAt_derivWeierstrassPExcept, tendsto_one_add_cpow_exp_of_tendsto, HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re, betaIntegral_recurrence, deriv_Gamma_add_one, HurwitzZeta.expZeta_one_sub, PeriodPair.mem_lattice, ModularGroup.denom_apply, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, NumberField.InfiniteAdeleRing.mixedEmbedding_eq_algebraMap_comp, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, arg_mul_cos_add_sin_mul_I_eq_toIocMod, arg_cos_add_sin_mul_I_sub, equivRealProdCLM_symm_apply, mellinInv_eq_fourierIntegralInv, norm_add_eq, cot_series_rep', norm_log_sub_logTaylor_le, Real.Angle.coe_toCircle, integerComplement.add_coe_int_mem, hasDerivAt_Gammaℝ_one, gelfandStarTransform_symm_apply, circleMap_pi_div_two, Quaternion.coeComplex_add, partialGamma_add_one, cos_add_sin_mul_I_pow, mellinInv_eq_fourierInv, fourier_gaussian_innerProductSpace', HurwitzZeta.hasSum_nat_hurwitzZetaEven, arg_mul_cos_add_sin_mul_I, integral_cpow, HurwitzZeta.hurwitzZeta_neg_nat, norm_log_one_add_sub_self_le, GaussianFourier.integrable_cexp_neg_mul_sum_add, AnalyticOnNhd.eq_const_add_im_mul_I_of_re_eq_const, ZetaAsymptotics.tendsto_Gamma_term_aux, normSq_add, inner_map_polarization', EisensteinSeries.linear_right_summable, LSeries_eq_mul_integral_of_nonneg, PeriodPair.periodic_derivWeierstrassP, EisensteinSeries.summable_linear_left_mul_linear_left, tendsto_one_add_div_pow_exp, log_ofReal_mul, norm_add_eq_iff, tendsto_logDeriv_euler_sin_div, EisensteinSeries.div_linear_zpow_differentiableOn, EulerSine.sin_pi_mul_eq, exp_periodic, tan_sub', deriv_Gamma_nat, ofReal_add, ProbabilityTheory.complexMGF_gaussianReal, add_re, HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one, hasDerivAt_jacobiTheta₂_fst, arg_one_add_mem_Ioo, HurwitzZeta.cosZeta_eq, PeriodPair.weierstrassPExcept_def, cos_add_pi, tendsto_norm_tan_atTop, hasDerivAt_Gammaℂ_one, riemannZeta_neg_two_mul_nat_add_one, sin_add_two_pi, ModularForm.eta_q_eq_cexp, HasDerivWithinAt.cpow, tendsto_nat_mul_log_one_add_of_tendsto, arg_mul_cos_add_sin_mul_I_coe_angle, EisensteinSeries.q_expansion_riemannZeta, SlashInvariantFormClass.periodic_comp_ofComplex, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, integral_boundary_rect_of_hasFDerivAt_real_off_countable, ZMod.LFunction_eq_completed_div_gammaFactor_odd, tan_add', one_add_cpow_hasFPowerSeriesAt_zero, jacobiTheta₂'_add_right, NumberField.mixedEmbedding.normAtPlace_add_le, cos_add_sin_I, hasSum_cauchyPowerSeries_integral, sin_eq_sin_iff, equivRealProdAddHom_symm_apply_re, HasProdLocallyUniformlyOn_euler_sin_prod, equivRealProdAddHom_symm_apply_im, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, one_div_sub_sq_hasFPowerSeriesOnBall_zero, EisensteinSeries.linear_left_summable, log_sub_self_isBigO, cosh_sq, LSeriesHasSum.add, GaussianInt.toComplex_add, hasSum_two_pi_I_cauchyPowerSeries_integral, cos_int_mul_two_pi_add_pi, eqOn_iteratedDeriv_cotTerm, exp_list_sum, tendsto_euler_sin_prod, normSq_ofReal_add_I_mul_sqrt_one_sub, GaussianInt.toComplex_def, ProbabilityTheory.hasDerivAt_complexMGF, tsum_exp_neg_quadratic, ofReal_comp_add, OnePoint.IsZeroAt.add, log_mul_eq_add_log_iff, cot_pi_eq_exp_ratio, EisensteinSeries.auxbound1, two_cosh, cot_series_rep, sin_antiperiodic, cosh_two_mul, cos_add, norm_mul_cos_add_sin_mul_I, PeriodPair.weierstrassP_add_coe, sin_eq_one_iff, equivRealProdLm_symm_apply, add_conj, hasDerivAt_logTaylor, ModularGroup.tendsto_normSq_coprime_pair, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, LSeriesSummable.add, ModularForm.slash_action_eq'_iff, cos_periodic, LSeries.abscissaOfAbsConv_add_le, EisensteinSeries.linear_inv_isBigO_right_add, integral_cexp_quadratic, GammaSeq_add_one_left, Real.circleAverage_eq_circleAverage_zero_one, orthonormalBasisOneI_repr_symm_apply, HurwitzZeta.sinZeta_two_mul_nat_add_one', tan_add_pi, Function.Periodic.cuspFunction_add, RCLike.complexLinearIsometryEquiv_symm_apply, sin_sq_add_cos_sq, sinh_add, HurwitzZeta.hasSum_nat_completedSinZeta, LSeries_eq_mul_integral, EisensteinSeries.isBigO_linear_add_const_vec, Gammaℂ_add_one, cos_add_pi_div_two, EisensteinSeries.auxbound2, Polynomial.X_pow_sub_X_sub_one_irreducible_aux, qExpansion_add, cexp_neg_quadratic_isLittleO_abs_rpow_cocompact, EisensteinSeries.qExpansion_identity_pnat, zeta_eq_tsum_one_div_nat_add_one_cpow, LSeries.term_add_apply, mem_slitPlane_of_norm_lt_one, EisensteinSeries.r_mul_max_le, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', Gamma_mul_Gamma_eq_betaIntegral, TemperedDistribution.smulLeftCLM_add, tendsto_one_add_div_cpow_exp, Polynomial.Chebyshev.U_complex_cosh, cos_add_cos, jacobiTheta₂'_add_left', cuspFunction_add, cos_sq_add_sin_sq, sin_periodic, RCLike.complexRingEquiv_symm_apply, GammaSeq_eq_betaIntegral_of_re_pos, hasSum_arctan_aux, EisensteinSeries.tendsto_zero_inv_linear, circleAverage_log_norm_add_const_eq_posLog, norm_add_mul_I, equivRealProdAddHom_apply, DirichletCharacter.deriv_LFunctionTrivChar₁_apply_of_ne_one, LSeries.term_add, DirichletCharacter.Odd.gammaFactor_def, cos_add_mul_I, DirichletCharacter.norm_LFunction_product_ge_one, EisensteinSeries.summable_linear_sub_mul_linear_add, measurableEquivPi_symm_apply, PeriodPair.latticeEquiv_symm_apply, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, EisensteinSeries.linear_isTheta_left, cos_add_two_pi, ZMod.LFunction_neg_two_mul_nat_add_one, multipliable_sineTerm, LindemannWeierstrass.integral_exp_mul_eval, UpperHalfPlane.coe_specialLinearGroup_apply, cotTerm_identity, GammaAux_recurrence1, LSeries_add, sin_add, wedgeIntegral_add_wedgeIntegral_eq, HurwitzZeta.sinZeta_two_mul_nat_add_one, integral_boundary_rect_eq_zero_of_differentiable_on_off_countable, sqrt_eq_real_add_ite, Polynomial.Chebyshev.S_two_mul_complex_cosh, log_mul, completedRiemannZeta₀_one, sin_eq, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, normSq_add_mul_I, GaussianFourier.integral_cexp_neg_mul_sum_add, EisensteinSeries.linear_isTheta_right_add, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, sin_eq_two_mul_tan_half_div_one_add_tan_half_sq, LSeries.tendsto_cpow_mul_atTop, EisensteinSeries.tendsto_double_sum_S_act, integrable_pow_mul_norm_one_add_mul_inv, sin_add_int_mul_two_pi, HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc, differentiableAt_Gamma_nat_add_one, HurwitzZeta.hurwitzZeta_one_sub, tan_add_int_mul_pi, OnePoint.IsBoundedAt.add, Gammaℝ_one_sub_mul_Gammaℝ_one_add, tan_add_nat_mul_pi, one_add_cpow_hasFPowerSeriesOnBall_zero, Gammaℝ_add_two, cexp_neg_quadratic_isLittleO_rpow_atTop, integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real, euler_sineTerm_tprod, cpow_add, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add, Polynomial.Chebyshev.U_complex_cos, sin_add_sin, LinearIsometry.im_apply_eq_im, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, ProbabilityTheory.complexMGF_id_gaussianReal, logDeriv_prod_sineTerm_eq_sum_cotTerm, ModularForm.coe_add, hasDerivAt_Gamma_one, HadamardThreeLines.eventuallyle, GammaSeq_mul, ProbabilityTheory.hasDerivAt_iteratedDeriv_complexMGF, sin_add_mul_I, EisensteinSeries.qExpansion_identity, GaussianFourier.integral_cexp_neg_sum_mul_add, tendsto_one_add_pow_exp_of_tendsto, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, log_sub_logTaylor_isBigO, cosh_add, UpperHalfPlane.coe_vadd, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, summable_log_one_add_of_summable, cos_sub_cos, inner_map_polarization, exp_eq_exp_re_mul_sin_add_cos, jacobiTheta_eq_tsum_nat, RCLike.norm_to_complex, Function.Periodic.qParam_left_inv_mod_period, Orientation.kahler_apply_apply, hasDerivAt_ofReal_cpow_const', hasDerivAt_Gamma_one_half, PeriodPair.derivWeierstrassPExcept_add_coe, GammaIntegral_add_one, DirichletCharacter.Even.LFunction_neg_two_mul_nat_add_one, Gamma_mul_Gamma_add_half, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, RCLike.to_complex_nonneg_iff, integral_Ioi_cpow_of_lt, DirichletCharacter.LFunctionTrivChar_isBigO_near_one_horizontal, cot_eq_exp_ratio, LSeries.pow_mul_term_eq, SlashInvariantForm.coe_add, cos_sq, hasSum_taylorSeries_log, norm_exp_mul_exp_add_exp_neg_le_of_abs_im_le, polarCoord_symm_apply, tan_sq_div_one_add_tan_sq, PeriodPair.periodic_weierstrassP, GaussianFourier.integrable_cexp_neg_sum_mul_add, EisensteinSeries.linear_isTheta_right, EisensteinSeries.summable_right_one_div_linear_sub_one_div_linear_succ, cos_eq_neg_one_iff, EisensteinSeries.summable_linear_right_add_one_mul_linear_right, jacobiTheta₂_add_left, log_mul_ofReal, jacobiTheta₂_add_left', riemannZeta_neg_nat_eq_bernoulli', re_add_im, one_add_cpow_hasFPowerSeriesOnBall_zero, betaIntegral_eval_nat_add_one_right, PeriodPair.mul_ω₁_add_mul_ω₂_mem_lattice, tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero, norm_log_one_add_half_le_self, hasMellin_cpow_Ioc, integral_boundary_rect_of_differentiableOn_real, integral_mul_cpow_one_add_sq, HadamardThreeLines.scale_id_mem_verticalStrip_of_mem_verticalStrip, integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn, SlashInvariantForm.slash_action_eqn', LSeries_hasDerivAt, integrable_cexp_quadratic, jacobiTheta₂_add_right, inv_one_add_tan_sq, cos_nat_mul_two_pi_add_pi, exp_mul_I_antiperiodic, exp_mul_I, riemannZeta_neg_nat_eq_bernoulli, DirichletCharacter.LFunction_isBigO_horizontal, isometryOfOrthonormal_symm_apply, arg_mul_cos_add_sin_mul_I_sub, NumberField.InfiniteAdeleRing.ringEquiv_mixedSpace_apply, tendsto_euler_sin_prod', multipliableUniformlyOn_euler_sin_prod_on_compact, sin_sub_sin, logTaylor_succ, HurwitzZeta.hasSum_int_completedSinZeta, eqOn_iteratedDerivWithin_cotTerm_integerComplement, MellinConvergent.cpow_smul, HurwitzZeta.hasSum_nat_hurwitzZetaOdd_of_mem_Icc, jacobiTheta₂'_add_left, tan_periodic, integrable_cexp_quadratic', logDeriv_sineTerm_eq_cotTerm, sin_add_nat_mul_two_pi, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, exp_mul_I_periodic, ModularForm.add_apply, cfcHom_eq_of_isStarNormal, mellin_cpow_smul, betaIntegral_eq_Gamma_mul_div, cos_eq_cos_iff, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, inv_Gammaℝ_two_sub, add_intCast_mem_integerComplement, EisensteinSeries.summand_bound, HadamardThreeLines.scale_id_mem_verticalClosedStrip_of_mem_verticalClosedStrip, cos_add_int_mul_two_pi, continuousFunctionalCalculus_map_id, PeriodPair.derivWeierstrassPExcept_def, arg_cos_add_sin_mul_I_eq_toIocMod, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, LSeries.hasDerivAt_term, LSeries_eq_mul_integral', log_exp_exists, Summable.hasSumUniformlyOn_log_one_add, horizontalSegment_eq, EisensteinSeries.D2_mul, CuspForm.coe_add, EulerSine.integral_cos_mul_cos_pow_even, ModularGroup.smul_eq_lcRow0_add, Gammaℝ_mul_Gammaℝ_add_one, fourierIntegral_gaussian_pi', PeriodPair.derivWeierstrassP_add_coe, HurwitzZeta.hurwitzZetaEven_eq, AnalyticOnNhd.eq_re_add_const_mul_I_of_re_eq_const, jacobiTheta_two_add, cpow_ofReal, sin_add_pi, RCLike.complexLinearIsometryEquiv_apply, cosh_add_sinh, PeriodPair.weierstrassPExcept_add, multipliable_one_add_of_summable, hasDerivAt_Gamma_nat, sin_add_pi_div_two, Summable.tendstoUniformlyOn_tsum_nat_log_one_add, exp_bound_sq, EisensteinSeries.linear_inv_isBigO_right, two_cos, lim_eq_lim_im_add_lim_re, rectangle_eq_convexHull, CuspForm.add_apply, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
instCoeReal 📖	CompOp	—
instCommSemiring 📖	CompOp	292 math math: ArithmeticFunction.vonMangoldt.residueClass_eq, norm_conj, log_conj_eq_ite, Orientation.kahler_map_complex, HasFDerivWithinAt.ccosh, ModularForm.smul_slash, jacobiTheta₂_conj, IsStarNormal.instNonUnitalContinuousFunctionalCalculus, sinh_conj, IsSelfAdjoint.quasispectrumRestricts, HasFDerivWithinAt.ccos, SchwartzMap.integral_sesq_fourier_fourier, Ideal.toCharacterSpace_apply_eq_zero_of_mem, lim_conj, HasStrictFDerivAt.csin, ofRealCLM_coe, kahler, areaForm, re_eq_add_conj, CliffordAlgebraComplex.ofComplex_conj, CStarModule.innerₛₗ_apply, has_pointwise_sum_fourier_series_of_summable, real_algHom_eq_id_or_conj, fderivWithin_csinh, conj_inv, Unitization.real_cfcₙ_eq_cfc_inr, star_def, nndist_self_conj, MeasureTheory.charFun_neg, StarAlgebra.elemental.characterSpaceToSpectrum_coe, Circle.ofConjDivSelf_coe, conjLIE_apply, Circle.starRingEnd_addChar, cosh_conj, LSeriesSummable.smul, oangle, UnitAddTorus.mFourierSubalgebra_coe, ZMod.dft_mul_const, CStarModule.innerSL_apply, cfc_complex_eq_real, UnitAddTorus.mFourierSubalgebra_closure_eq_top, isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts, HasFDerivWithinAt.csin, CliffordAlgebraComplex.ofComplex_toComplex, CliffordAlgebraComplex.toComplex_comp_ofComplex, gelfandStarTransform_naturality, isCauSeq_conj, gelfandStarTransform_apply_apply, jacobiTheta₂'_conj, Polynomial.toAddCircle_X_pow_eq_fourier, cos_conj, fderivWithin_ccosh, conj_rootsOfUnity, EisensteinSeries.E2_slash_action, Unitization.instStarOrderedRing, SchwartzMap.integral_sesq_fourier_eq, isometry_conj, summable_conj, log_inv_eq_ite, mul_conj', hasStrictFDerivAt_cpow', liftAux_apply, Orientation.kahler_swap, IsSelfAdjoint.coe_mem_spectrum_complex, UnitDisc.coe_conj, BoundedContinuousFunction.mem_charPoly, spectrum.gelfandTransform_eq, BoundedContinuousFunction.charAlgHom_apply, lift_apply, Unitary.norm_sub_one_lt_two_iff, real_linearMap_map_smul_complex, Gamma_conj, cuspFunction_smul, ringHom_eq_id_or_conj_of_continuous, dist_self_conj, gelfandStarTransform_symm_apply, Polynomial.toAddCircle.integrable, fourier_neg', LSeries_smul, IsSelfAdjoint.val_re_map_spectrum, hasStrictFDerivAt_cpow, normSq_add, CStarAlgebra.instNonnegSpectrumClassComplexNonUnital, HasFDerivWithinAt.cpow, HasStrictFDerivAt.const_cpow, HasFDerivWithinAt.csinh, IsSelfAdjoint.isConnected_spectrum_compl, UpperHalfPlane.sigma_J, VonNeumannAlgebra.centralizer_centralizer', inv_cpow_eq_ite', StarAlgebra.elemental.bijective_characterSpaceToSpectrum, CliffordAlgebraComplex.ofComplex_I, fderivWithin_ccos, HasStrictFDerivAt.clog, liftAux_neg_I, spectrum.map_pow, cfcₙ_real_eq_complex, ModularForm.instIsScalarTowerComplex, UnitAddTorus.mFourier_neg, normSq_sub, Circle.coe_inv_eq_conj, HasStrictFDerivAt.ccos, det_conjAe, sub_conj, Orientation.kahler_neg_orientation, IsSelfAdjoint.cfc_arg, conj_im, liftAux_I, fourierSubalgebra_separatesPoints, continuous_conj, hasFDerivAt_cpow, conj_cpow, LSeriesHasSum.smul, VonNeumannAlgebra.mem_carrier, HasFDerivAt.ccos, NumberField.IsCMField.complexEmbedding_complexConj, range_liftAux, HasFDerivAt.clog, inv_cpow_eq_ite, liftAux_apply_I, StarSubalgebra.spectrum_eq, inv_def, cfcHom_real_eq_restrict, arg_conj, inner_map_complex, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, AnalyticAt.harmonicAt_conj, Algebra.norm_complex_eq, HasFDerivAt.ccosh, GaussianInt.toComplex_star, conjAe_coe, conj_mul', inr_comp_cfcₙHom_eq_cfcₙAux, sin_conj, conj_eq_iff_re, QuasispectrumRestricts.real_iff, conj_cpow_eq_ite, normSq_conj, add_conj, tanh_conj, PositiveLinearMap.gnsStarAlgHom_apply, spectrum.nonempty, isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts, UpperHalfPlane.J_smul, instNonemptyElemWeakDualComplexCharacterSpaceOfNontrivial, spectrum_subset_unitary_of_mem_unitary, AddChar.starComp_apply, fderiv_ccos, CliffordAlgebraComplex.equiv_symm_apply, cot_conj, SchwartzMap.integral_sesq_fourierIntegral_eq, NormedRing.algEquivComplexOfComplete_symm_apply, Unitary.spectrum_subset_slitPlane_iff_norm_lt_two, UpperHalfPlane.petersson_symm, CliffordAlgebraComplex.toComplex_ι, linearEquiv_det_conjAe, LSeriesSummable.smul_iff, HasFDerivWithinAt.clog, fourierSubalgebra_coe, Polynomial.toAddCircle_monomial_eq_smul_fourier, inner, UpperHalfPlane.coe_J_smul, NumberField.conj_basisMatrix, Algebra.norm_complex_apply, Unitization.nnreal_cfcₙ_eq_cfc_inr, UnitAddTorus.hasSum_mFourier_series_of_summable, UnitAddTorus.mFourierSubalgebra_separatesPoints, AddChar.starComp_eq_inv, HasFDerivAt.csin, WStarAlgebra.exists_predual, Orientation.kahler_rotation_left, ModularFormClass.hasSum_qExpansion_of_abs_lt, hasSum_conj, NormedRing.algEquivComplexOfComplete_apply, IsStarNormal.instContinuousFunctionalCalculus, LSeries.term_smul_apply, CuspForm.instIsScalarTowerComplex, dist_conj_conj, Polynomial.fourierCoeff_toAddCircle_natCast, conjCLE_apply, arg_conj_coe_angle, nndist_conj_self, UpperHalfPlane.cosh_half_dist, CliffordAlgebraComplex.equiv_apply, Function.Periodic.cuspFunction_smul, HasFDerivWithinAt.const_cpow, StarSubalgebra.mem_spectrum_iff, lift_symm_apply_coe, UpperHalfPlane.exp_half_dist, uniformContinuous_ringHom_eq_id_or_conj, fourier_neg, fderivWithin_csin, Algebra.trace_complex_apply, WeakDual.CharacterSpace.mem_spectrum_iff_exists, conj_re, im_eq_sub_conj, conj_ofNat, LSeries.mul_delta_eq_smul_delta, UnitAddTorus.hasSum_prod_mFourierCoeff, hasSum_conj', hasSum_fourier_series_of_summable, ZMod.dft_const_mul, exp_conj, unitary_iff_isStarNormal_and_spectrum_subset_unitary, UpperHalfPlane.sinh_half_dist_add_dist, fderiv_ccosh, LinearIsometry.im_apply_eq_im, HasStrictFDerivAt.cpow, mul_conj, IsStarNormal.instNonUnitalIsometricContinuousFunctionalCalculus, nndist_conj_comm, VonNeumannAlgebra.coe_toStarSubalgebra, StarSubalgebra.coe_isUnit, conj_natCast, nndist_conj_conj, instContinuousMapUniqueHom, Orientation.areaForm_map_complex, cpow_conj, fourierSubalgebra_closure_eq_top, selfAdjoint.val_re_map_spectrum, Unitization.complex_cfcₙ_eq_cfc_inr, conj_eq_iff_real, toMatrix_conjAe, ModularFormClass.hasSum_qExpansion_of_norm_lt, normSq_eq_conj_mul_self, conj_eq_iff_im, LSeries.delta_mul_eq_smul_delta, dist_conj_comm, GammaIntegral_conj, CliffordAlgebraComplex.toComplex_involute, CliffordAlgebraComplex.toComplex_ofComplex, Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv, UnitAddTorus.hasSum_mFourier_series_apply_of_summable, cfc_real_eq_complex, spectrum.exists_nnnorm_eq_spectralRadius, conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj, unitary.norm_sub_one_lt_two_iff, HurwitzZeta.jacobiTheta₂''_conj, adjoin_I, LSeries.term_smul, UpperHalfPlane.σ_conj, fderiv_csinh, NumberField.canonicalEmbedding.conj_apply, conj_ofReal, NormedAlgebra.Complex.nonempty_algEquiv, HasFDerivAt.csinh, CliffordAlgebraComplex.ofComplex_comp_toComplex, HasStrictFDerivAt.ccosh, Orientation.oangle_map_complex, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, Polynomial.toAddCircle_X_eq_fourier_one, inv_eq_conj, conj_neg_I, TemperedDistribution.smulLeftCLM_smul, ofRealAm_coe, UpperHalfPlane.tanh_half_dist, fderiv_csin, Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero, HasFDerivAt.cpow, conj_tsum, NumberField.ComplexEmbedding.conjugate_coe_eq, IsStarNormal.instIsometricContinuousFunctionalCalculus, CStarAlgebra.instNonnegSpectrumClassComplexUnital, Unitary.argSelfAdjoint_coe, tan_conj, WeakDual.CharacterSpace.exists_apply_eq_zero, LinearMap.isSymmetric_iff_inner_map_self_real, spectrum.map_polynomial_aeval, Polynomial.toAddCircle_C_eq_smul_fourier_zero, BoundedContinuousFunction.separatesPoints_charPoly, HasStrictFDerivAt.csinh, UnitDisc.coe_star, qExpansion_smul, cfcHom_eq_of_isStarNormal, Polynomial.fourierCoeff_toAddCircle, ModularFormClass.hasSum_qExpansion, log_conj, summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp, conj_I, continuousFunctionalCalculus_map_id, NormedAlgebra.Complex.exists_norm_sub_smul_one_eq_zero, CFC.complex_exp_eq_normedSpace_exp, dist_conj_self, cfcₙ_complex_eq_real, WeakDual.CharacterSpace.instStarHomClass, conjCLE_coe, BoundedContinuousFunction.char_mem_charPoly, HasFDerivAt.const_cpow, exp_bound_sq, Unitization.cfcₙ_eq_cfc_inr, Polynomial.fourierCoeff_toAddCircle_eq_zero_of_lt_zero, unitary.spectrum_subset_slitPlane_iff_norm_lt_two
instDecidableEq 📖	CompOp	13 math math: PeriodPair.hasSum_derivWeierstrassPExcept, PeriodPair.ite_eq_one_sub_sq_mul_weierstrassP, differentiableOn_update_limUnder_of_isLittleO, differentiableOn_update_limUnder_of_bddAbove, PeriodPair.hasSumLocallyUniformly_weierstrassPExcept, Polynomial.eq_centerMass_of_eval_derivative_eq_zero, HurwitzZeta.differentiableAt_update_of_residue, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, Polynomial.sum_derivRootWeight_pos, PeriodPair.hasSum_weierstrassPExcept, cpow_def, Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv, differentiableOn_update_limUnder_insert_of_isLittleO
instDivInvMonoid 📖	CompOp	458 math math: HurwitzZeta.hasSum_int_hurwitzZetaOdd, cpow_intCast, hasSum_mellin_pi_mul_sq', cos_eq_tsum, HasDerivWithinAt.clog_real, HurwitzZeta.differentiableAt_hurwitzZetaEven_sub_one_div, EulerSine.integral_cos_mul_cos_pow, log_I, div_ofReal_re, ProbabilityTheory.isGaussian_iff_gaussian_charFun, completedZeta_eq_tsum_of_one_lt_re, HurwitzZeta.differentiableAt_hurwitzZeta_sub_one_div, Gammaℝ_def, sin_pi_div_two_sub, PeriodPair.hasSumLocallyUniformly_weierstrassP, HurwitzZeta.hasSum_int_hurwitzZetaEven, re_eq_add_conj, PeriodPair.derivWeierstrassPExcept_sub, ProbabilityTheory.hasGaussianLaw_iff_charFunDual_map_eq, HurwitzZeta.cosZeta_two_mul_nat', PeriodPair.ω₁_div_two_notMem_lattice, UpperHalfPlane.re_smul, ZMod.LFunction_one_sub, DirichletCharacter.continuousOn_neg_logDeriv_LFunction_of_nontriv, UpperHalfPlane.int_div_mem_integerComplement, tsum_exp_neg_mul_int_sq, GaussianFourier.integral_cexp_neg_mul_sq_norm, div_ofNat_im, arg_div_self, angle_div_right_eq_angle_mul_left, hasSum_one_div_nat_pow_mul_fourier, cos_eq_zero_iff, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, mellin_div_const, WeakFEPair.symm_Λ₀_eq, HasProdUniformlyOn_sineTerm_prod_on_compact, HurwitzZeta.cosZeta_two_mul_nat, Circle.ofConjDivSelf_coe, sin_eq_neg_one_iff, UpperHalfPlane.specialLinearGroup_apply, riemannZeta_two, hasStrictDerivAt_tan, fourierIntegral_gaussian_innerProductSpace', fourier_gaussian_pi', jacobiTheta₂_functional_equation, ProbabilityTheory.hasGaussianLaw_iff_charFun_map_eq, GaussianFourier.integral_cexp_neg_mul_sq_norm_add, UpperHalfPlane.moebius_im, tan_pi_div_two_sub, DirichletCharacter.LSeries_twist_vonMangoldt_eq, HurwitzZeta.jacobiTheta₂'_functional_equation', HurwitzZeta.hurwitzZetaEven_def_of_ne_or_ne, cos_pi_div_two, hasSum_nat_jacobiTheta, PeriodPair.hasSum_derivWeierstrassPExcept, digamma_one_half, ZMod.stdAddChar_coe, ModularForm.tsum_logDeriv_eta_q, EisensteinSeries.summable_left_one_div_linear_sub_one_div_linear, HurwitzZeta.oddKernel_def', cos_eq_two_mul_tan_half_div_one_sub_tan_half_sq, fourier_gaussian_innerProductSpace, DirichletCharacter.IsPrimitive.completedLFunction_one_sub, GaussianInt.toComplex_div_re, EisensteinSeries.E2_slash_action, mem_rootsOfUnity, UpperHalfPlane.glPos_smul_def, sin_bound, PeriodPair.ite_eq_one_sub_sq_mul_weierstrassP, HurwitzZeta.hasSum_nat_sinZeta, fourierCoeff_bernoulli_eq, EulerSine.integral_cos_mul_cos_pow_aux, LSeries.term_def, riemannZeta_one_sub, PeriodPair.ω₂_div_two_notMem_lattice, PeriodPair.hasSum_derivWeierstrassP, integral_exp_mul_complex_Ioi, PeriodPair.weierstrassP_bound, PeriodPair.hasSumLocallyUniformly_derivWeierstrassP, ofReal_zpow, betaIntegral_eval_one_right, cauchyPowerSeries_apply, HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re, HurwitzZeta.hurwitzZetaEven_apply_zero, HurwitzZeta.expZeta_one_sub, HurwitzZeta.cosZeta_apply_zero, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, fourierIntegral_gaussian_pi, hasSum_mellin, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, integral_exp_mul_complex, cos_eq_tsum', cot_series_rep', hasDerivAt_Gammaℝ_one, HadamardThreeLines.mem_verticalClosedStrip_of_scale_id_mem_verticalClosedStrip, cpow_sub, div_ofNat, riemannZeta_four, ZMod.LFunction_eq_completed_div_gammaFactor_even, HurwitzZeta.hasSum_nat_completedCosZeta, tendsto_sub_mul_tsum_nat_cpow, sin_sub_pi_div_two, fourier_gaussian_innerProductSpace', HurwitzZeta.hasSum_nat_hurwitzZetaEven, LSeries.term_of_ne_zero, div_ratCast_re, integral_cpow, Real.tendsto_integral_gaussian_smul', HurwitzZeta.hurwitzZeta_neg_nat, EisensteinSeries.tendsto_zero_inv_linear_sub, exp_neg_pi_div_two_mul_I, tendsto_riemannZeta_sub_one_div, ZetaAsymptotics.tendsto_Gamma_term_aux, exp_nsmul', inner_map_polarization', HasStrictDerivAt.clog_real, EisensteinSeries.linear_right_summable, EisensteinSeries.D2_S, Polynomial.int_cyclotomic_rw, tan_int_mul_pi_div_two, EisensteinSeries.G2_S_transform, div_im, tendsto_one_add_div_pow_exp, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, tendsto_logDeriv_euler_sin_div, EisensteinSeries.div_linear_zpow_differentiableOn, ProbabilityTheory.IsGaussian.charFun_eq', logDeriv_sin_div_eq_cot, HurwitzZeta.cosZeta_one_sub, EulerSine.sin_pi_mul_eq, jacobiTheta_S_smul, zeta_eq_tsum_one_div_nat_cpow, isPrimitiveRoot_iff, MellinConvergent.div_const, WeakFEPair.Λ₀_eq, circleIntegral.integral_sub_zpow_of_undef, UpperHalfPlane.denom_cocycle, norm_exp_sub_sum_le_norm_mul_exp, tan_sub', UpperHalfPlane.coe_smul_of_det_pos, ProbabilityTheory.complexMGF_gaussianReal, isPrimitiveRoot_exp_of_isCoprime, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div, HadamardThreeLines.interpStrip_scale, HurwitzZeta.cosZeta_eq, GaussianInt.toComplex_im_div, PeriodPair.weierstrassPExcept_def, HurwitzZeta.hasSum_int_cosZeta, DirichletCharacter.LFunction_eq_completed_div_gammaFactor, isCauSeq_norm_exp, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, tendsto_norm_tan_atTop, ProbabilityTheory.IsGaussian.charFunDual_eq_of_forall_strongDual_eq_zero, hasDerivAt_Gammaℂ_one, hasSum_sin, EisensteinSeries.q_expansion_riemannZeta, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, fourier_coe_apply, ZMod.LFunction_eq_completed_div_gammaFactor_odd, eisSummand_of_gammaSet_eq_divIntMap, tan_add', HurwitzZeta.hasSum_nat_cosZeta, tanh_eq_sinh_div_cosh, HurwitzZeta.completedHurwitzZetaEven_eq, angle_eq_abs_arg, isBigO_riemannZeta_sub_one_div, ProbabilityTheory.isGaussian_iff_charFun_eq, fourierIntegral_eq_half_sub_half_period_translate, HasProdLocallyUniformlyOn_euler_sin_prod, has_antideriv_at_fourier_neg, UpperHalfPlane.im_smul, HurwitzZeta.hasSum_expZeta_of_one_lt_re, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, one_div_sub_sq_hasFPowerSeriesOnBall_zero, EisensteinSeries.linear_left_summable, PeriodPair.hasSumLocallyUniformly_weierstrassPExcept, UpperHalfPlane.coe_smul, div_intCast, completedRiemannZeta_eq, ModularForm.logDeriv_one_sub_mul_cexp_comp, hasSum_two_pi_I_cauchyPowerSeries_integral, eqOn_iteratedDeriv_cotTerm, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, hasDerivAt_fourier_neg, tendsto_euler_sin_prod, ZMod.completedLFunction_eq, EulerSine.integral_sin_mul_sin_mul_cos_pow_eq, HasDerivAt.clog, ProbabilityTheory.IsGaussian.charFunDual_eq, tsum_exp_neg_quadratic, riemannZeta_zero, cot_pi_eq_exp_ratio, DiffContOnCl.circleAverage_smul_div, cosh_eq_tsum, HurwitzZeta.hurwitzZetaEven_one_sub, hasSum_mellin_pi_mul, hasMellin_one_Ioc, cot_series_rep, bernoulliFourierCoeff_eq, isPrimitiveRoot_exp_of_coprime, ModularForm.logDeriv_eta_eq_E2, sin_eq_one_iff, norm_zpow, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, ModularForm.slash_action_eq'_iff, exp_pi_div_two_mul_I, GaussianInt.toComplex_re_div, integral_cexp_quadratic, HasDerivWithinAt.clog, GammaSeq_add_one_left, SlashInvariantForm.slash_action_eqn'', HurwitzZeta.sinZeta_two_mul_nat_add_one', tan_eq_sin_div_cos, cos_bound, div_re, hasDerivAt_fourier, one_div_sub_pow_hasFPowerSeriesOnBall_zero, isPrimitiveRoot_exp, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, ModularForm.logDeriv_one_sub_cexp, arg_eq_arg_iff, angle_div_left_eq_angle_mul_right, HurwitzZeta.hasSum_nat_completedSinZeta, UpperHalfPlane.contMDiff_denom_zpow, circleIntegral_div_sub_of_differentiable_on_off_countable, Gammaℂ_add_one, cos_add_pi_div_two, norm_div, fourierIntegral_gaussian_innerProductSpace, ProbabilityTheory.HasGaussianLaw.charFun_map_eq, ArithmeticFunction.LSeries_vonMangoldt_eq, EisensteinSeries.qExpansion_identity_pnat, zeta_eq_tsum_one_div_nat_add_one_cpow, ModularForm.SL_slash_def, div_ofReal, PeriodPair.hasSum_weierstrassP, tendsto_one_add_div_cpow_exp, hasSum_cos', cos_add_cos, arg_div_coe_angle, jacobiTheta₂'_functional_equation, riemannZeta_one, exp_sub_sum_range_succ_isLittleO_pow, normSq_div, log_div_self, sin_pi_div_two, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, rotationOf_coe, cos_sub_pi_div_two, hasSum_arctan_aux, deriv_tan, circleMap_zero_div, EisensteinSeries.tendsto_zero_inv_linear, LSeries.term_of_ne_zero', tendsto_logDeriv_euler_cot_sub, mellin_comp_rpow, div_ofNat_re, exp_sub_sum_range_isBigO_pow, isOpenQuotientMap_zpow_compl_zero, IsExpCmpFilter.isLittleO_zpow_mul_exp, hasSum_mellin_pi_mul₀, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, HurwitzZeta.differentiableAt_update_of_residue, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, SlashInvariantForm.slash_action_eqn_SL'', exp_bound, circleIntegral.integral_sub_zpow_of_ne, MellinConvergent.comp_rpow, div_intCast_re, circleAverage_sub_sub_inv_smul_of_differentiable_on, EisensteinSeries.q_expansion_bernoulli, UpperHalfPlane.coe_specialLinearGroup_apply, cotTerm_identity, GammaAux_recurrence1, HurwitzZeta.sinZeta_two_mul_nat_add_one, EulerSine.tendsto_integral_cos_pow_mul_div, circleIntegrable_sub_zpow_iff, completedRiemannZeta₀_one, fourierCoeffOn_of_hasDerivAt_Ioo, HurwitzZeta.hurwitzZetaOdd_eq, div_ratCast_im, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, EulerSine.antideriv_cos_comp_const_mul, GaussianFourier.integral_cexp_neg_mul_sum_add, ModularFormClass.qExpansion_coeff_eq_circleIntegral, ProbabilityTheory.isGaussian_iff_charFunDual_eq, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, sin_eq_two_mul_tan_half_div_one_add_tan_half_sq, fourierIntegral_gaussian, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, EisensteinSeries.tendsto_double_sum_S_act, sinh_eq_tsum, HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc, im_eq_sub_conj, EulerSine.antideriv_sin_comp_const_mul, div_natCast_im, cot_eq_cos_div_sin, HurwitzZeta.hurwitzZeta_one_sub, integral_gaussian_sq_complex, div_I, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun, sin_eq_tsum, Gammaℝ_one_sub_mul_Gammaℝ_one_add, SlashInvariantForm.slash_S_apply, tan_eq_zero_iff, Gammaℝ_add_two, hasSum_taylorSeries_neg_log, euler_sineTerm_tprod, exp_bound', ofReal_div, sin_add_sin, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, ZMod.LFunction_apply_zero_of_even, HurwitzZeta.completedCosZeta_eq, ProbabilityTheory.complexMGF_id_gaussianReal, summable_one_div_nat_cpow, cpow_int_mul', PeriodPair.hasSum_weierstrassPExcept, GammaSeq_mul, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, EisensteinSeries.qExpansion_identity, GaussianFourier.integral_cexp_neg_sum_mul_add, sameRay_iff_arg_div_eq_zero, cos_pi_div_two_sub, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar_val, HurwitzZeta.hasSum_int_sinZeta, ZMod.toCircle_intCast, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, cos_sub_cos, ZMod.toCircle_natCast, inner_map_polarization, div_ofReal_im, cpow_mul_int, hasDerivAt_ofReal_cpow_const', hasDerivAt_Gamma_one_half, ModularForm.slash_def, Real.tendsto_integral_gaussian_smul, bernoulliFourierCoeff_recurrence, exp_sub, Gamma_mul_Gamma_add_half, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, integral_Ioi_cpow_of_lt, HurwitzZeta.sinKernel_def, DirichletCharacter.LFunctionTrivChar_isBigO_near_one_horizontal, cot_eq_exp_ratio, fourier_coe_apply', cos_sq, hasSum_taylorSeries_log, ProbabilityTheory.IsGaussian.charFunDual_eq_of_integral_eq_zero, hasDerivAt_tan, ProbabilityTheory.IsGaussian.charFun_eq, integral_gaussian_complex, tan_sq_div_one_add_tan_sq, summable_bernoulli_fourier, ZMod.toCircle_apply, hasSum_sinh, integral_exp_mul_complex_Iic, norm_exp_sub_sum_le_exp_norm_sub_sum, completedRiemannZeta_one, Gammaℝ_div_Gammaℝ_one_sub, EisensteinSeries.summable_right_one_div_linear_sub_one_div_linear_succ, hasSum_sin', Gamma_mul_Gamma_one_sub, log_neg_I, hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, cpow_int_mul, riemannZeta_neg_nat_eq_bernoulli', sin_eq_tsum', ModularForm.SL_slash_apply, betaIntegral_eval_nat_add_one_right, ZMod.LFunction_residue_one, riemannZeta_def_of_ne_zero, EisensteinSeries.eisSummand_SL2_apply, div_intCast_im, hasMellin_cpow_Ioc, integral_mul_cpow_one_add_sq, SlashInvariantForm.slash_action_eqn', div_natCast, one_div_one_sub_hasFPowerSeriesOnBall_zero, fourierCoeffOn_of_hasDerivAt, HurwitzZeta.tendsto_hurwitzZeta_sub_one_div_nhds_one, inv_Gammaℝ_one_sub, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, hasSum_arctan, HurwitzZeta.hasSum_int_completedCosZeta, hasSum_cosh, riemannZeta_neg_nat_eq_bernoulli, HurwitzZeta.sinZeta_eq, ProbabilityTheory.IsGaussian.charFunDual_eq', riemannZeta_two_mul_nat, UpperHalfPlane.im_pnat_div_pos, ModularForm.logDeriv_qParam, tendsto_euler_sin_prod', ProbabilityTheory.charFun_gaussianReal, sin_sub_sin, hasSum_cos, logTaylor_succ, HurwitzZeta.hurwitzZetaOdd_one_sub, HurwitzZeta.hasSum_int_completedSinZeta, eqOn_iteratedDerivWithin_cotTerm_integerComplement, zeta_nat_eq_tsum_of_gt_one, Circle.coe_zpow, HurwitzZeta.hasSum_nat_hurwitzZetaOdd_of_mem_Icc, isCauSeq_exp, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, circleAverage_sub_sub_inv_smul_of_differentiable_on_off_countable, HurwitzZeta.hasSum_int_completedHurwitzZetaEven, WeakFEPair.f_modif_aux2, UpperHalfPlane.mdifferentiable_denom_zpow, betaIntegral_eq_Gamma_mul_div, integral_cpow_mul_exp_neg_mul_Ioi, Circle.coe_div, integral_cos_mul_complex, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, HurwitzZeta.sinZeta_one_sub, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq, summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp, inv_Gammaℝ_two_sub, Gamma_one_half_eq, fourierCoeffOn_of_hasDeriv_right, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_nhds_right, exp_int_mul, arg_zpow_coe_angle, HasDerivAt.clog_real, PeriodPair.derivWeierstrassPExcept_def, div_natCast_re, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, fourier_gaussian_pi, div_ratCast, GaussianInt.normSq_div_sub_div_lt_one, DirichletCharacter.continuousOn_neg_logDeriv_LFunctionTrivChar₁, EulerSine.integral_cos_mul_cos_pow_even, Gammaℂ_one, one_div_sub_hasFPowerSeriesOnBall_zero, ModularGroup.smul_eq_lcRow0_add, fourierIntegral_gaussian_pi', tan_two_mul, HurwitzZeta.hurwitzZetaEven_eq, integral_gaussian_complex_Ioi, hasSum_mellin_pi_mul_sq, GaussianInt.toComplex_div_im, PeriodPair.weierstrassPExcept_add, HasStrictDerivAt.clog, tan_eq_one_sub_tan_half_sq_div_one_add_tan_half_sq, sin_add_pi_div_two, ModularForm.slash_apply, HurwitzZeta.tendsto_hurwitzZetaEven_sub_one_div_nhds_one, circleMap_zero_zpow, UpperHalfPlane.petersson_slash, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
instField 📖	CompOp	94 math math: ArithmeticFunction.vonMangoldt.residueClass_eq, PeriodPair.hasFPowerSeriesAt_weierstrassPExcept, NumberField.mixedEmbedding.norm_eq_sup'_normAtPlace, DirichletCharacter.LSeries_eulerProduct_tprod, Polynomial.mahlerMeasure_eq_leadingCoeff_mul_prod_roots, one_add_cpow_hasFPowerSeriesAt_zero, DirichletCharacter.LSeries_modOne_eq, DirichletCharacter.LSeriesSummable_of_one_lt_re, Polynomial.one_le_prod_max_one_norm_roots, re_balance, im_expect, DirichletCharacter.LSeries_twist_vonMangoldt_eq, DirichletCharacter.LFunction_eq_LSeries, DirichletCharacter.norm_LSeries_product_ge_one, DirichletCharacter.deriv_LFunction_eq_deriv_LSeries, DirichletCharacter.LFunction_changeLevel, DirichletCharacter.mul_delta, PeriodPair.hasFPowerSeriesAt_derivWeierstrassPExcept, ArithmeticFunction.vonMangoldt.residueClass_apply, DirichletCharacter.LSeriesSummable_mul, NumberField.canonicalEmbedding.integerLattice.inter_ball_finite, NumberField.InfinitePlace.card_filter_mk_eq, DirichletCharacter.LSeries_changeLevel, re_comp_balance, AddChar.expect_apply_eq_zero_iff_ne_zero, ofReal_balance, Polynomial.int_cyclotomic_rw, DirichletCharacter.LSeries.mul_mu_eq_one, NumberField.InfinitePlace.ComplexEmbedding.conjugate_sign, Polynomial.prod_max_one_norm_roots_le_mahlerMeasure_of_one_le_leadingCoeff, one_add_cpow_hasFPowerSeriesAt_zero, Polynomial.rootSet_derivative_subset_convexHull_rootSet, one_div_sub_sq_hasFPowerSeriesOnBall_zero, DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow, cfcHom_real_eq_restrict, IsSelfAdjoint.spectrumRestricts, Polynomial.logMahlerMeasure_eq_log_leadingCoeff_add_sum_log_roots, Polynomial.eq_centerMass_of_eval_derivative_eq_zero, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, SpectrumRestricts.real_iff, NumberField.canonicalEmbedding.nnnorm_eq, DirichletCharacter.LSeriesSummable_twist_vonMangoldt, DirichletCharacter.convolution_twist_vonMangoldt, ofReal_comp_balance, NumberField.discr_eq_basisMatrix_det_sq, im_balance, ofReal_expect, one_div_sub_pow_hasFPowerSeriesOnBall_zero, DirichletCharacter.LSeriesSummable_iff, DirichletCharacter.absicssaOfAbsConv_eq_one, MulChar.star_apply', im_comp_balance, NumberField.inverse_basisMatrix_mulVec_eq_repr, NumberField.InfinitePlace.card_real_embeddings, isOpenQuotientMap_zpow_compl_zero, DirichletCharacter.norm_LFunction_product_ge_one, cfcₙHom_real_eq_restrict, Polynomial.sum_derivRootWeight_pos, DirichletCharacter.LSeries_eulerProduct_hasProd, card_primitiveRoots, tsum_dirichletSummand, DirichletCharacter.LSeries_eulerProduct_exp_log, NumberField.canonicalEmbedding.norm_le_iff, hasFPowerSeriesOnBall_cuspFunction, MulChar.star_eq_inv, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, one_add_cpow_hasFPowerSeriesOnBall_zero, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, DirichletCharacter.not_LSeriesSummable_at_one, card_rootsOfUnity, rank_rat_complex, Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv, re_expect, DirichletCharacter.delta_mul, isIntegral_rat_I, DirichletCharacter.convolution_mul_moebius, one_add_cpow_hasFPowerSeriesOnBall_zero, one_div_one_sub_hasFPowerSeriesOnBall_zero, isAlgClosed, NumberField.canonicalEmbedding_eq_basisMatrix_mulVec, Circle.toUnits_apply, DirichletCharacter.modZero_eq_delta, DirichletCharacter.fourierTransform_eq_gaussSum_mulShift, AddChar.expect_apply_eq_ite, DirichletCharacter.LSeries_eulerProduct, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, NumberField.InfinitePlace.card_complex_embeddings, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, NumberField.house_eq_sup', nonempty_linearEquiv_real, ofReal_choose, one_div_sub_hasFPowerSeriesOnBall_zero, Polynomial.int_cyclotomic_spec
instInhabited 📖	CompOp	1 math math: Function.Periodic.cuspFunction_zero_eq_limUnder_nhds_ne
instIntCast 📖	CompOp	146 math math: Polynomial.Chebyshev.S_two_mul_complex_cos, cpow_intCast, EisensteinSeries.linear_inv_isBigO_left, HurwitzZeta.hasSum_int_cosKernel₀, exp_eq_exp_iff_exists_int, Int.cast_complex_isTheta_cast_real, GaussianInt.toComplex_def', mem_integerComplement_iff, cos_eq_one_iff, UpperHalfPlane.int_div_mem_integerComplement, tsum_exp_neg_mul_int_sq, EisensteinSeries.summand_bound_of_mem_verticalStrip, two_mul_riemannZeta_eq_tsum_int_inv_pow_of_even, cos_eq_zero_iff, sin_eq_neg_one_iff, sin_int_mul_pi, sin_eq_zero_iff, ArithmeticFunction.LSeries_zeta_mul_Lseries_moebius, ZMod.stdAddChar_coe, ModularGroup.coe_apply_complex, tan_int_mul_pi, EisensteinSeries.summable_left_one_div_linear_sub_one_div_linear, sin_int_mul_two_pi_sub, ModularGroup.coe_T_zpow_smul_eq, ArithmeticFunction.not_LSeriesSummable_moebius_at_one, fourierCoeff_bernoulli_eq, circleMap_eq_circleMap_iff, tan_eq_zero_iff', PeriodPair.mem_lattice, ModularGroup.denom_apply, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, integerComplement.add_coe_int_mem, HurwitzZeta.hasSum_int_cosKernel, EisensteinSeries.tendsto_zero_inv_linear_sub, EisensteinSeries.linear_right_summable, tan_int_mul_pi_div_two, EisensteinSeries.summable_linear_left_mul_linear_left, cos_int_mul_two_pi_sub, EisensteinSeries.div_linear_zpow_differentiableOn, norm_exp_mul_sq_le, log_exp_eq_sub_toIocDiv, ofReal_intCast, isPrimitiveRoot_exp_of_isCoprime, isOpen_compl_range_intCast, DirichletCharacter.LSeries.mul_mu_eq_one, Polynomial.Chebyshev.T_complex_cosh, Polynomial.Chebyshev.C_two_mul_complex_cosh, HurwitzZeta.hasSum_int_cosZeta, tendsto_norm_tan_atTop, intCast_re, fourier_coe_apply, tan_sub_int_mul_pi, sin_eq_sin_iff, has_antideriv_at_fourier_neg, EisensteinSeries.linear_left_summable, div_intCast, cos_int_mul_two_pi_add_pi, hasDerivAt_fourier_neg, GaussianInt.toComplex_def, tsum_exp_neg_quadratic, HurwitzZeta.hasSum_int_sinKernel, bernoulliFourierCoeff_eq, sin_eq_one_iff, ModularGroup.tendsto_normSq_coprime_pair, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, ModularForm.slash_action_eq'_iff, normSq_intCast, EisensteinSeries.linear_inv_isBigO_right_add, NumberField.discr_eq_basisMatrix_det_sq, hasDerivAt_fourier, EisensteinSeries.isBigO_linear_add_const_vec, EisensteinSeries.qExpansion_identity_pnat, EisensteinSeries.r_mul_max_le, Polynomial.Chebyshev.U_complex_cosh, norm_int_of_nonneg, EisensteinSeries.tendsto_zero_inv_linear, tan_int_mul_pi_sub, Polynomial.Chebyshev.T_complex_cos, EisensteinSeries.summable_linear_sub_mul_linear_add, PeriodPair.latticeEquiv_symm_apply, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, EisensteinSeries.linear_isTheta_left, cos_int_mul_two_pi_sub_pi, LSeries_one_mul_Lseries_moebius, div_intCast_re, Polynomial.Chebyshev.S_two_mul_complex_cosh, fourierCoeffOn_of_hasDerivAt_Ioo, integerComplement_eq, ArithmeticFunction.LSeriesSummable_moebius_iff, EisensteinSeries.linear_isTheta_right_add, EisensteinSeries.tendsto_double_sum_S_act, sin_add_int_mul_two_pi, tan_add_int_mul_pi, ModularForm.coe_intCast, tan_eq_zero_iff, Polynomial.Chebyshev.U_complex_cos, Polynomial.Chebyshev.C_two_mul_complex_cos, cpow_int_mul', EisensteinSeries.qExpansion_identity, HurwitzZeta.hasSum_int_sinZeta, ZMod.toCircle_intCast, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, Function.Periodic.qParam_left_inv_mod_period, nnnorm_intCast, cpow_mul_int, sin_sub_int_mul_two_pi, bernoulliFourierCoeff_recurrence, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, fourier_coe_apply', summable_bernoulli_fourier, EisensteinSeries.linear_isTheta_right, EisensteinSeries.summable_right_one_div_linear_sub_one_div_linear_succ, cos_eq_neg_one_iff, DirichletCharacter.convolution_mul_moebius, EisensteinSeries.summable_linear_right_add_one_mul_linear_right, hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, cpow_int_mul, div_intCast_im, fourierCoeffOn_of_hasDerivAt, exp_eq_one_iff, HurwitzZeta.hasSum_int_completedCosZeta, intCast_im, HurwitzZeta.hasSum_int_completedSinZeta, norm_intCast, cos_eq_cos_iff, add_intCast_mem_integerComplement, integerComplement.mem_iff, EisensteinSeries.summand_bound, cos_int_mul_two_pi, cos_add_int_mul_two_pi, fourierCoeffOn_of_hasDeriv_right, exp_int_mul, exp_int_mul_two_pi_mul_I, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, log_exp_exists, SlashInvariantForm.coe_intCast, GaussianInt.intCast_complex_norm, ModularGroup.smul_eq_lcRow0_add, cos_sub_int_mul_two_pi, isometry_intCast, EisensteinSeries.linear_inv_isBigO_right, isClosed_range_intCast, ArithmeticFunction.abscissaOfAbsConv_moebius, closedEmbedding_intCast, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
instInv 📖	CompOp	152 math math: ArithmeticFunction.vonMangoldt.residueClass_eq, norm_log_one_sub_inv_add_logTaylor_neg_le, EisensteinSeries.linear_inv_isBigO_left, circleTransformDeriv_eq, DirichletCharacter.LSeries_eulerProduct_tprod, digamma_apply_add_one, inv_I, hasSum_taylorSeries_of_entire, conj_inv, ZMod.invDFT_apply, circleIntegral.integral_sub_inv_of_mem_ball, two_mul_riemannZeta_eq_tsum_int_inv_pow_of_even, WeakFEPair.symm_Λ₀_eq, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, hasSum_taylorSeries_on_ball, sum_cauchyPowerSeries_eq_integral, tan_pi_div_two_sub, taylorSeries_eq_of_entire, cpow_neg_one, norm_one_add_mul_inv_le, circleMap_zero_inv, DifferentiableOn.circleIntegral_sub_inv_smul, integral_circleTransform, cot_inv_eq_tan, one_div_Gamma_eq_self_mul_one_div_Gamma_add_one, Polynomial.logMahlerMeasure_of_degree_eq_one, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, taylorSeries_eq_on_ball', log_inv_eq_ite, circleIntegral.integral_sub_inv_smul_sub_smul, UpperHalfPlane.mdifferentiable_inv_denom, ValueDistribution.proximity_inv, cauchyPowerSeries_apply, DiffContOnCl.circleIntegral_sub_inv_smul, ArithmeticFunction.vonMangoldt.residueClass_apply, log_inv, Real.circleAverage_eq_circleIntegral, UpperHalfPlane.contMDiff_inv_denom, EisensteinSeries.linear_right_summable, ValueDistribution.characteristic_sub_characteristic_inv_of_ne_zero, EisensteinSeries.G2_S_transform, EisensteinSeries.summable_linear_left_mul_linear_left, integral_mul_cexp_neg_mul_sq, ValueDistribution.characteristic_sub_characteristic_inv_at_zero, inv_cpow_eq_ite', HasStrictFDerivAt.clog, cpow_neg, inv_re, WeakFEPair.h_feq', UpperHalfPlane.modular_S_smul, circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const, EisensteinSeries.q_expansion_riemannZeta, eisSummand_of_gammaSet_eq_divIntMap, hasSum_cauchyPowerSeries_integral, pow_cpow_ofNat_inv, differentiable_Gammaℝ_inv, HasFDerivAt.clog, exp_neg, inv_cpow_eq_ite, one_div_sub_sq_hasFPowerSeriesOnBall_zero, EisensteinSeries.linear_left_summable, inv_cpow, inv_def, ZMod.invDFT_def', two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, ValueDistribution.characteristic_sub_characteristic_inv_le, hasSum_two_pi_I_cauchyPowerSeries_integral, hasStrictDerivAt_log, ModularFormClass.qExpansion_coeff, inv_zero, ZMod.invDFT_apply', inv_natCast_cpow_ofReal_pos, abs_arg_inv, ValueDistribution.characteristic_sub_characteristic_inv, EisensteinSeries.linear_inv_isBigO_right_add, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, cpow_inv_two_im_eq_sqrt, cpow_inv_two_re, ValueDistribution.isBigO_characteristic_sub_characteristic_inv, NumberField.mixedEmbedding.det_matrixToStdBasis, one_div_sub_pow_hasFPowerSeriesOnBall_zero, EisensteinSeries.isBigO_linear_add_const_vec, hasSum_taylorSeries_on_emetric_ball, hasFPowerSeriesOn_cauchy_integral, circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable, HasFDerivWithinAt.clog, normSq_inv, continuousOn_prod_circle_transform_function, pow_cpow_nat_inv, riemannZeta_eulerProduct, EisensteinSeries.summable_linear_sub_mul_linear_add, taylorSeries_eq_on_eball, DirichletCharacter.LSeries_eulerProduct_hasProd, ValueDistribution.proximity_zero_of_complexValued, differentiable_one_div_Gamma, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, ModularFormClass.qExpansion_coeff_eq_circleIntegral, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, EisensteinSeries.tendsto_double_sum_S_act, arg_inv, integrable_pow_mul_norm_one_add_mul_inv, DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, taylorSeries_eq_on_emetric_ball, Gammaℝ_one_sub_mul_Gammaℝ_one_add, SlashInvariantForm.slash_S_apply, Circle.coe_inv, taylorSeries_eq_on_ball, log_eq_integral, cpow_nat_inv_pow, inv_im, circleIntegrable_sub_inv_iff, norm_cpow_inv_nat, mul_inv_cancel, UpperHalfPlane.im_inv_neg_coe_pos, taylorSeries_eq_on_eball', ZMod.invDFT_def, norm_log_one_sub_inv_sub_self_le, cpow_inv_two_im_eq_neg_sqrt, PeriodPair.hasSum_sumInvPow, DirichletCharacter.LFunctionTrivChar_residue_one, circleIntegral.integral_sub_center_inv, Polynomial.logMahlerMeasure_C_mul_X_add_C, hasStrictFDerivAt_log_real, Real.circleAverage_zero_one_congr_inv, EisensteinSeries.summable_linear_right_add_one_mul_linear_right, riemannZeta_eulerProduct_tprod, arg_inv_coe_angle, inv_eq_conj, two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, inv_one_add_tan_sq, inv_Gammaℝ_one_sub, taylorSeries_eq_of_entire', hasDerivAt_log_sub_logTaylor, abs_cpow_inv_two_im, hasDerivAt_log, riemannZeta_eulerProduct_hasProd, DirichletCharacter.LSeries_eulerProduct, log_inv_eq_integral, cpow_ofNat_inv_pow, ofReal_inv, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, inv_Gammaℝ_two_sub, sq_cpow_two_inv, tan_inv_eq_cot, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, continuous_circleMap_inv, ValueDistribution.proximity_sub_proximity_inv_eq_circleAverage, taylorSeries_eq_on_emetric_ball', one_div_sub_hasFPowerSeriesOnBall_zero, continuousOn_one_add_mul_inv, EisensteinSeries.linear_inv_isBigO_right, hasSum_taylorSeries_on_eball
instMul 📖	CompOp	979 math math: ArithmeticFunction.vonMangoldt.residueClass_eq, HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip_eps, ModularForm.coe_mul, Real.hasDerivAt_fourier, Polynomial.Chebyshev.S_two_mul_complex_cos, verticalSegment_eq, Orientation.kahler_map_complex, RCLike.sqrt_eq_ite, cos_eq_tsum, ModularForm.mul_slash_SL2, ProbabilityTheory.integrable_cexp_mul_of_re_mem_integrableExpSet, EulerSine.integral_cos_mul_cos_pow, HasDerivAt.ccos, mul_im, tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero, log_I, one_add_tan_sq_mul_cos_sq_eq_one, re_mul_ofReal, GaussianInt.toComplex_mul, ProbabilityTheory.isGaussian_iff_gaussian_charFun, nnnorm_exp_I_mul_ofReal, log_neg_one, completedZeta_eq_tsum_of_one_lt_re, EisensteinSeries.linear_inv_isBigO_left, GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I, PeriodPair.hasFPowerSeriesAt_weierstrassPExcept, Real.hasFDerivAt_fourierChar_neg_bilinear_left, cos_three_mul, EisensteinSeries.hasSum_e2Summand_symmetricIcc, mk_eq_add_mul_I, AddDissociated.randomisation, Gammaℝ_def, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, HurwitzZeta.hasSum_int_cosKernel₀, exp_eq_exp_iff_exists_int, kahler, areaForm, Real.tsum_eq_tsum_fourier, Real.fourierIntegralInv_eq', cos_mul_I, ModularForm.mul_slash, ZMod.LFunction_modOne_eq, cos_sub_sin_I, tan_nat_mul_pi_sub, GaussianInt.toComplex_def', integral_boundary_rect_eq_zero_of_differentiableOn, MeasureTheory.charFunDual_apply, DirichletCharacter.LSeries_eulerProduct_tprod, PeriodPair.iteratedDeriv_weierstrassPExcept_self, LSeries.term_convolution', tsum_eisSummand_eq_riemannZeta_mul_eisensteinSeries, isLittleO_exp_neg_mul_sq_cocompact, ProbabilityTheory.hasGaussianLaw_iff_charFunDual_map_eq, HurwitzZeta.cosZeta_two_mul_nat', real_smul, summableLocallyUniformlyOn_iteratedDerivWithin_cexp, cos_eq_one_iff, equivRealProdAddHom_symm_apply, ValueDistribution.characteristic_mul_top_le, ModularForm.eta_q_eq_pow, HurwitzZeta.sinZeta_neg_two_mul_nat_sub_one, Orientation.kahler_comp_rightAngleRotation', ZMod.LFunction_one_sub, tsum_exp_neg_mul_int_sq, Real.fderiv_fourierChar_neg_bilinear_right_apply, exp_add_mul_I, deriv_cpow_const, Quaternion.coeComplex_mul, SchwartzMap.lineDerivOp_fourier_eq, arg_cos_add_sin_mul_I_coe_angle, EisensteinSeries.summand_bound_of_mem_verticalStrip, circleMap_neg_pi_div_two, GaussianFourier.integral_cexp_neg_mul_sq_norm, ArithmeticFunction.LSeriesHasSum_mul, Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable, angle_exp_one, circleIntegral.integral_sub_inv_of_mem_ball, ValueDistribution.proximity_zero_mul_le, angle_div_right_eq_angle_mul_left, hasSum_one_div_nat_pow_mul_fourier, two_mul_riemannZeta_eq_tsum_int_inv_pow_of_even, betaIntegral_scaled, cos_eq_zero_iff, cos_sub, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, fourierCoeffOn.const_mul, cos_add_nat_mul_two_pi, deriv_csinh, TemperedDistribution.lineDerivOp_fourier_eq, HasProdUniformlyOn_sineTerm_prod_on_compact, mul_re, HurwitzZeta.cosZeta_two_mul_nat, norm_eq_one_iff', sin_eq_neg_one_iff, UpperHalfPlane.specialLinearGroup_apply, cpow_mul_ofReal_nonneg, ProbabilityTheory.integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, HurwitzZeta.cosZeta_neg_two_mul_nat_add_one, fourierIntegral_gaussian_innerProductSpace', sqrt_I, fourier_gaussian_pi', oangle, TemperedDistribution.smulLeftCLM_compL_smulLeftCLM, iteratedDerivWithin_tsum_cexp_eq, PeriodPair.iteratedDeriv_derivWeierstrassPExcept_self, jacobiTheta₂_functional_equation, ProbabilityTheory.hasGaussianLaw_iff_charFun_map_eq, cosh_three_mul, HasStrictDerivAt.cpow, sin_int_mul_pi, GaussianFourier.integral_cexp_neg_mul_sq_norm_add, norm_cexp_neg_mul_sq, sin_eq_zero_iff, sum_cauchyPowerSeries_eq_integral, DirichletCharacter.LSeries_twist_vonMangoldt_eq, Real.norm_exp_I_mul_ofReal_sub_one_le, Real.vector_fourierIntegral_eq_integral_exp_smul, HurwitzZeta.evenKernel_def, Gamma_add_one, Real.fourier_real_eq_integral_exp_smul, norm_one_add_mul_inv_le, tendsto_tsum_powerSeries_nhdsWithin_stolzCone, HurwitzZeta.jacobiTheta₂'_functional_equation', norm_cos_add_sin_mul_I, angle_mul_left, hasSum_nat_jacobiTheta, Real.fourier_deriv, DirichletCharacter.norm_LSeries_product_ge_one, exp_add, ArithmeticFunction.LSeries_zeta_mul_Lseries_moebius, rotation_apply, digamma_one_half, ZMod.stdAddChar_coe, DifferentiableOn.circleIntegral_sub_inv_smul, gelfandStarTransform_naturality, gelfandStarTransform_apply_apply, HarmonicAt.analyticAt_complex_partial, Real.nnnorm_exp_I_mul_ofReal_sub_one_le, WeakFEPair.h_feq, arg_mul_eq_add_arg_iff, ZMod.completedLFunction_def_even, sin_nat_mul_pi, tan_int_mul_pi, ModularForm.tsum_logDeriv_eta_q, EisensteinSeries.summable_left_one_div_linear_sub_one_div_linear, integral_circleTransform, HasDerivAt.cpow, DirichletCharacter.LFunction_changeLevel, HurwitzZeta.oddKernel_def', fourier_gaussian_innerProductSpace, tan_nat_mul_pi, DirichletCharacter.IsPrimitive.completedLFunction_one_sub, isPrimitiveRoot_exp_rat, VectorFourier.fourierPowSMulRight_eq_comp, EisensteinSeries.E2_slash_action, mem_rootsOfUnity, MeasureTheory.charFun_apply, one_div_Gamma_eq_self_mul_one_div_Gamma_add_one, sin_int_mul_two_pi_sub, cos_nat_mul_two_pi_sub_pi, Polynomial.logMahlerMeasure_of_degree_eq_one, hasDerivAt_GammaIntegral, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, deriv_ofReal_cpow_const, LSeries_iteratedDeriv, two_sinh, PeriodPair.ite_eq_one_sub_sq_mul_weierstrassP, exp_antiperiodic, MeasureTheory.charFun_apply_real, taylorSeries_eq_on_ball', UnitDisc.coe_smul_circle, fourierCoeff_bernoulli_eq, exp_add_pi_mul_I, log_exp_eq_re_add_toIocMod, UpperHalfPlane.denom_cocycle_σ, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace, circleMap_eq_circleMap_iff, ZMod.LFunction_def_odd, EulerSine.integral_cos_mul_cos_pow_aux, SchwartzMap.fourier_lineDerivOp_eq, equivRealProd_symm_apply, normSq_mul, UpperHalfPlane.re_add_im, DirichletCharacter.mul_delta, TemperedDistribution.fourierInv_lineDerivOp_eq, mul_conj', RCLike.complexRingEquiv_apply, HasStrictDerivAt.csin, hasStrictFDerivAt_cpow', arg_cos_add_sin_mul_I, cpow_ofNat_mul', cos_eq_iff_quadratic, riemannZeta_one_sub, sinh_three_mul, integral_exp_mul_complex_Ioi, ProbabilityTheory.integrable_rpow_abs_mul_cexp_of_re_mem_interior_integrableExpSet, tan_eq_zero_iff', cos_nat_mul_two_pi, natCast_mul_natCast_cpow, NumberField.tendsto_sub_one_mul_dedekindZeta_nhdsGT, PeriodPair.hasFPowerSeriesAt_derivWeierstrassPExcept, cauchyPowerSeries_apply, Orientation.rotation_map_complex, ofReal_mul', betaIntegral_recurrence, HasDerivWithinAt.csinh, sin_sub_nat_mul_two_pi, DiffContOnCl.circleIntegral_sub_inv_smul, deriv_Gamma_add_one, LSeriesHasSum.convolution, ofReal_cpow_of_nonpos, sin_nat_mul_two_pi_sub, sin_two_mul, HurwitzZeta.expZeta_one_sub, PeriodPair.mem_lattice, BoundedContinuousFunction.mem_charPoly, ModularGroup.denom_apply, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, BoundedContinuousFunction.charAlgHom_apply, ArithmeticFunction.vonMangoldt.residueClass_apply, NumberField.InfiniteAdeleRing.mixedEmbedding_eq_algebraMap_comp, fourierIntegral_gaussian_pi, ProbabilityTheory.complexMGF_mul_I, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, Real.circleAverage_eq_circleIntegral, norm_exp_I_mul_ofReal_sub_one, integral_exp_mul_complex, MeasureTheory.charFunDual_prod, arg_mul_cos_add_sin_mul_I_eq_toIocMod, arg_cos_add_sin_mul_I_sub, equivRealProdCLM_symm_apply, cos_eq_tsum', mellinInv_eq_fourierIntegralInv, cot_series_rep', MeasureTheory.charFun_eq_prod_iff, DirichletCharacter.LSeriesSummable_mul, exists_norm_eq_mul_self, Real.Angle.coe_toCircle, hasDerivAt_Gammaℝ_one, HasDerivWithinAt.const_cpow, gelfandStarTransform_symm_apply, circleMap_pi_div_two, cos_two_mul, HurwitzZeta.hasSum_nat_completedCosZeta, tendsto_sub_mul_tsum_nat_cpow, cos_sub_two_pi, partialGamma_add_one, sin_sub_two_pi, DirichletCharacter.LSeries_changeLevel, cos_add_sin_mul_I_pow, mellinInv_eq_fourierInv, norm_eq_one_iff, exp_ofReal_mul_I_re, fourier_gaussian_innerProductSpace', arg_mul_cos_add_sin_mul_I, HurwitzZeta.hasSum_int_cosKernel, ZMod.completedLFunction_def_odd, sqrt_neg_of_nonneg, LSeries_smul, IsExpCmpFilter.isLittleO_exp_cpow, Real.tendsto_integral_gaussian_smul', HurwitzZeta.hurwitzZeta_neg_nat, EisensteinSeries.tendsto_zero_inv_linear_sub, exp_neg_pi_div_two_mul_I, GaussianFourier.integrable_cexp_neg_mul_sum_add, hasStrictFDerivAt_cpow, AnalyticOnNhd.eq_const_add_im_mul_I_of_re_eq_const, ZetaAsymptotics.tendsto_Gamma_term_aux, exp_nsmul', HasDerivAt.csinh, normSq_add, inner_map_polarization', EisensteinSeries.linear_right_summable, MeasureTheory.charFunDual_eq_prod_iff, EisensteinSeries.D2_S, Polynomial.int_cyclotomic_rw, ZMod.LFunction_neg_two_mul_nat_sub_one, tan_int_mul_pi_div_two, UnitAddTorus.mFourier_add, LSeries_eq_mul_integral_of_nonneg, HasFDerivWithinAt.cpow, EisensteinSeries.G2_S_transform, angle_mul_right, EisensteinSeries.summable_linear_left_mul_linear_left, HurwitzZeta.oddKernel_def, HasStrictFDerivAt.const_cpow, isIntegral_exp_rat_mul_pi_mul_I, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, log_ofReal_mul, cos_int_mul_two_pi_sub, iteratedDeriv_odd_sin, HarmonicAt.differentiableAt_complex_partial, tendsto_logDeriv_euler_sin_div, EisensteinSeries.div_linear_zpow_differentiableOn, Real.fourierIntegral_iteratedDeriv, ProbabilityTheory.IsGaussian.charFun_eq', logDeriv_sin_div_eq_cot, HurwitzZeta.cosZeta_one_sub, EulerSine.sin_pi_mul_eq, integral_mul_cexp_neg_mul_sq, jacobiTheta_S_smul, isPrimitiveRoot_iff, exp_periodic, norm_exp_mul_sq_le, deriv_ccosh, exp_nat_mul_two_pi_mul_I, log_exp_eq_sub_toIocDiv, UpperHalfPlane.denom_cocycle, HasDerivAt.ccosh, tan_sub', ProbabilityTheory.complexMGF_id_mul_I, deriv_Gamma_nat, PeriodPair.weierstrassPExcept_eq_tsum, isAlgebraic_sin_rat_mul_pi, ProbabilityTheory.complexMGF_gaussianReal, UnitDisc.coe_closedBall_smul, isPrimitiveRoot_exp_of_isCoprime, Real.tsum_eq_tsum_fourier_of_rpow_decay_of_summable, Gammaℝ_eq_zero_iff, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, DirichletCharacter.LSeries.mul_mu_eq_one, HurwitzZeta.completedHurwitzZetaEven_residue_one, HasStrictDerivAt.csinh, HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one, Polynomial.Chebyshev.T_complex_cosh, Polynomial.Chebyshev.C_two_mul_complex_cosh, cpow_nat_mul, integral_exp_mul_I_eq_sinc, normSq_sub, HasDerivAt.csin, sin_two_pi, UpperHalfPlane.norm_exp_two_pi_I_lt_one, HurwitzZeta.hasSum_int_cosZeta, arg_exp_mul_I, WeakFEPair.h_feq', ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, tendsto_norm_tan_atTop, BoundedContinuousFunction.innerProbChar_apply, hasDerivAt_Gammaℂ_one, hasSum_sin, tendsto_self_mul_Gamma_nhds_zero, riemannZeta_neg_two_mul_nat_add_one, ValueDistribution.proximity_mul_zero_le, hasStrictDerivAt_const_cpow, sub_conj, sin_add_two_pi, EisensteinSeries.hasSum_e2Summand_symmetricIco, ModularForm.eta_q_eq_cexp, HasDerivWithinAt.cpow, Circle.coe_mul, arg_mul_cos_add_sin_mul_I_coe_angle, circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const, EisensteinSeries.q_expansion_riemannZeta, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, Real.fourierIntegral_real_eq_integral_exp_smul, fourier_coe_apply, Real.deriv_fourierIntegral, integral_exp_mul_I_eq_sin, HasStrictDerivAt.cexp, integral_boundary_rect_of_hasFDerivAt_real_off_countable, eisSummand_of_gammaSet_eq_divIntMap, tan_add', DirichletCharacter.LFunctionTrivChar_eq_mul_riemannZeta, cos_add_sin_I, hasSum_cauchyPowerSeries_integral, tan_sub_int_mul_pi, sin_eq_sin_iff, LSeries_convolution', LSeries.term_convolution, hasFDerivAt_cpow, ProbabilityTheory.isGaussian_iff_charFun_eq, LSeriesHasSum.smul, HasProdLocallyUniformlyOn_euler_sin_prod, has_antideriv_at_fourier_neg, HasDerivWithinAt.ccosh, HadamardThreeLines.interpStrip_eq_of_pos, HurwitzZeta.hasSum_expZeta_of_one_lt_re, sin_three_mul, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, EisensteinSeries.e2Summand_zero_eq_two_riemannZeta_two, PeriodPair.summable_weierstrassPSummand, one_div_sub_sq_hasFPowerSeriesOnBall_zero, DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow, EisensteinSeries.linear_left_summable, inv_def, ModularForm.logDeriv_one_sub_mul_cexp_comp, two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, inner_map_complex, PeriodPair.derivWeierstrassP_sq, ArithmeticFunction.LSeries_mul', MeasureTheory.charFun_conv, cos_sub_nat_mul_two_pi, cos_int_mul_two_pi_add_pi, mul_cpow_ofReal_nonneg, eqOn_iteratedDeriv_cotTerm, exp_list_sum, hasDerivAt_fourier_neg, SchwartzMap.fourierInv_lineDerivOp_eq, cos_two_mul', tendsto_euler_sin_prod, MeasureTheory.integral_charFun_Icc, ZMod.completedLFunction_eq, normSq_ofReal_add_I_mul_sqrt_one_sub, derivWithin_ccosh, exp_ofReal_mul_I_im, derivWithin_cexp, EulerSine.integral_sin_mul_sin_mul_cos_pow_eq, HurwitzZeta.hurwitzZeta_residue_one, GaussianInt.toComplex_def, ProbabilityTheory.IsGaussian.charFunDual_eq, tsum_exp_neg_quadratic, ModularFormClass.qExpansion_coeff, conj_mul', isAddQuotientCoveringMap_exp, log_mul_eq_add_log_iff, cot_pi_eq_exp_ratio, torusIntegral_const_mul, EisensteinSeries.auxbound1, isIntegral_two_mul_sin_rat_mul_pi, mul_I_re, HurwitzZeta.hurwitzZetaEven_one_sub, deriv_csin, ProbabilityTheory.indepFun_iff_charFunDual_prod', HurwitzZeta.hasSum_int_sinKernel, ofReal_comp_mul, ValueDistribution.proximity_mul_top_le, DirichletCharacter.LSeriesSummable_twist_vonMangoldt, tan_mul_cos, two_cosh, cot_series_rep, HurwitzZeta.completedCosZeta_residue_zero, HurwitzZeta.LSeriesHasSum_exp, DirichletCharacter.convolution_twist_vonMangoldt, ValueDistribution.characteristic_mul_zero_eventuallyLE, deriv_cexp, bernoulliFourierCoeff_eq, sinh_two_mul, ValueDistribution.characteristic_top_mul_le, cosh_two_mul, cos_add, norm_mul_cos_add_sin_mul_I, isPrimitiveRoot_exp_of_coprime, tan_sub_nat_mul_pi, abel_aux, ModularForm.logDeriv_eta_eq_E2, cosh_sub, TemperedDistribution.fourier_lineDerivOp_eq, sin_eq_one_iff, equivRealProdLm_symm_apply, MeasureTheory.charFunDual_map_const_add, MeasureTheory.charFunDual_map_add_const, hasDerivAt_logTaylor, ModularGroup.tendsto_normSq_coprime_pair, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, ModularForm.slash_action_eq'_iff, Real.fourier_iteratedDeriv, cos_periodic, ProbabilityTheory.indepFun_iff_charFun_prod, WeakFEPair.hf_zero, cpow_mul_nat, exp_pi_div_two_mul_I, EisensteinSeries.linear_inv_isBigO_right_add, VectorFourier.fourierSMulRight_apply, integral_cexp_quadratic, cuspFunction_mul, GammaSeq_add_one_left, Real.circleAverage_eq_circleAverage_zero_one, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, orthonormalBasisOneI_repr_symm_apply, SlashInvariantForm.slash_action_eqn'', HurwitzZeta.sinZeta_two_mul_nat_add_one', hasDerivAt_circleMap, Gammaℂ_def, norm_mul_exp_arg_mul_I, HasStrictDerivAt.cpow_const, NumberField.mixedEmbedding.det_matrixToStdBasis, isPrimitiveRoot_exp_rat_of_even_num, RCLike.complexLinearIsometryEquiv_symm_apply, iteratedDeriv_even_cos, hasDerivAt_fourier, sinh_add, ZMod.LFunction_def_even, one_div_sub_pow_hasFPowerSeriesOnBall_zero, isPrimitiveRoot_exp, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, ModularForm.logDeriv_one_sub_cexp, isIntegral_exp_neg_rat_mul_pi_mul_I, arg_eq_arg_iff, angle_div_left_eq_angle_mul_right, HurwitzZeta.hasSum_nat_completedSinZeta, circleMap_zero, integrableOn_exp_mul_complex_Iic, LSeries_eq_mul_integral, Real.fourierIntegral_eq', EisensteinSeries.isBigO_linear_add_const_vec, hasFPowerSeriesOn_cauchy_integral, circleIntegral_div_sub_of_differentiable_on_off_countable, contDiffOn_tsum_cexp, Gammaℂ_add_one, deriv_circleMap, EisensteinSeries.auxbound2, fourierIntegral_gaussian_innerProductSpace, ProbabilityTheory.HasGaussianLaw.charFun_map_eq, HasDerivWithinAt.csin, cexp_neg_quadratic_isLittleO_abs_rpow_cocompact, EisensteinSeries.qExpansion_identity_pnat, DirichletCharacter.IsPrimitive.fourierTransform_eq_inv_mul_gaussSum, circleMap_zero_mul, Real.iteratedDeriv_fourier, Orientation.kahler_rotation_left', ValueDistribution.characteristic_mul_zero_le, cot_pi_mul_contDiffWithinAt, EisensteinSeries.r_mul_max_le, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', ModularForm.SL_slash_def, Gamma_mul_Gamma_eq_betaIntegral, hasSum_cos', Polynomial.Chebyshev.U_complex_cosh, ProbabilityTheory.IndepFun.charFunDual_map_add_eq_mul, cos_add_cos, LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_and_nonneg, jacobiTheta₂'_add_left', inner, jacobiTheta₂'_functional_equation, riemannZeta_one, derivWithin_ccos, ProbabilityTheory.integrable_cexp_mul_of_re_mem_interior_integrableExpSet, natCast_cpow_natCast_mul, sin_periodic, RCLike.complexRingEquiv_symm_apply, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, GammaSeq_eq_betaIntegral_of_re_pos, UnitDisc.coe_mul, hasSum_arctan_aux, EisensteinSeries.tendsto_zero_inv_linear, deriv_const_cpow, cpow_ofNat_mul, UnitDisc.coe_smul_closedBall, norm_add_mul_I, DirichletCharacter.deriv_LFunctionTrivChar₁_apply_of_ne_one, ZMod.completedLFunction_const_mul, tan_int_mul_pi_sub, tendsto_logDeriv_euler_cot_sub, Polynomial.Chebyshev.T_complex_cos, cos_add_mul_I, DirichletCharacter.norm_LFunction_product_ge_one, IsExpCmpFilter.isLittleO_zpow_mul_exp, EisensteinSeries.summable_linear_sub_mul_linear_add, measurableEquivPi_symm_apply, Orientation.kahler_rightAngleRotation_left, PeriodPair.latticeEquiv_symm_apply, tanh_mul_I, betaIntegral_convergent, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, EisensteinSeries.linear_isTheta_left, enorm_exp_I_mul_ofReal, cos_int_mul_two_pi_sub_pi, PeriodPair.weierstrassPSeries_hasSum, SlashInvariantForm.slash_action_eqn_SL'', cos_add_two_pi, derivWithin_csin, NumberField.mixedEmbedding.unitSMul_smul, Real.hasDerivAt_fourierChar, ZMod.LFunction_neg_two_mul_nat_add_one, Orientation.kahler_rotation_left, LindemannWeierstrass.integral_exp_mul_eval, LSeries.iteratedDeriv_alternating, LSeries_one_mul_Lseries_moebius, exists_norm_mul_eq_self, DirichletCharacter.LSeries_eulerProduct_hasProd, PeriodPair.deriv_derivWeierstrassPExcept_self, Real.tendsto_integral_cexp_sq_smul, Real.enorm_exp_I_mul_ofReal_sub_one_le, HasStrictDerivAt.const_cpow, EisensteinSeries.q_expansion_bernoulli, UpperHalfPlane.coe_specialLinearGroup_apply, cotTerm_identity, deriv_ccos, HurwitzZeta.cosKernel_def, I_mul, LSeries.term_smul_apply, sin_add, SchwartzMap.integral_fourier_mul_eq, I_mul_re, wedgeIntegral_add_wedgeIntegral_eq, sqrt_neg_I, HurwitzZeta.sinZeta_two_mul_nat_add_one, integral_boundary_rect_eq_zero_of_differentiable_on_off_countable, sqrt_eq_real_add_ite, EulerSine.tendsto_integral_cos_pow_mul_div, Polynomial.Chebyshev.S_two_mul_complex_cosh, log_mul, completedRiemannZeta₀_one, HasDerivWithinAt.ccos, fourierCoeffOn_of_hasDerivAt_Ioo, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, SchwartzMap.lineDerivOp_fourierInv_eq, arg_mul_real, DirichletCharacter.LSeries_eulerProduct_exp_log, HasFDerivWithinAt.const_cpow, ZMod.completedLFunction_one_sub_odd, sin_eq, Real.fourierChar_apply, DirichletCharacter.Even.LFunction_neg_two_mul_nat, HadamardThreeLines.interpStrip_eq_of_mem_verticalStrip, ValueDistribution.characteristic_zero_mul_eventually_le, arg_mul, Gammaℝ_residue_zero, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, EulerSine.antideriv_cos_comp_const_mul, normSq_add_mul_I, Orientation.kahler_mul, GaussianFourier.integral_cexp_neg_mul_sum_add, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat_sub_one, LSeries.mul_delta, ModularFormClass.qExpansion_coeff_eq_circleIntegral, ProbabilityTheory.isGaussian_iff_charFunDual_eq, EisensteinSeries.linear_isTheta_right_add, sin_eq_two_mul_tan_half_div_one_add_tan_half_sq, fourierIntegral_gaussian, LSeries.tendsto_cpow_mul_atTop, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, EisensteinSeries.tendsto_double_sum_S_act, im_mul_ofReal, integrable_pow_mul_norm_one_add_mul_inv, sin_add_int_mul_two_pi, im_eq_sub_conj, EulerSine.antideriv_sin_comp_const_mul, HurwitzZeta.hurwitzZeta_one_sub, WeakFEPair.hf_modif_FE, integral_gaussian_sq_complex, MeasureTheory.charFunDual_prod', DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, exp_pi_mul_I, derivWithin_const_cpow, LSeries.mul_delta_eq_smul_delta, isPrimitiveRoot_exp_rat_of_odd_num, tan_add_int_mul_pi, hasDerivAt_ofReal_cpow_const, I_mul_I, UnitAddTorus.hasSum_prod_mFourierCoeff, isIntegral_two_mul_cos_rat_mul_pi, div_I, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun, sin_eq_tsum, Gammaℝ_one_sub_mul_Gammaℝ_one_add, tan_add_nat_mul_pi, fourierCoeff.const_mul, ArithmeticFunction.LSeries_mul, Real.tsum_eq_tsum_fourier_of_rpow_decay, cpow_nat_mul', SlashInvariantForm.slash_S_apply, exp_sub_pi_mul_I, SchwartzMap.fderivCLM_fourier_eq, LSeries.delta_mul, hasStrictDerivAt_cpow_const, tan_eq_zero_iff, Gammaℝ_add_two, GammaSeq_eq_approx_Gamma_integral, nnnorm_exp_ofReal_mul_I, HasDerivWithinAt.cexp, cexp_neg_quadratic_isLittleO_rpow_atTop, integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real, ProbabilityTheory.integrable_pow_abs_mul_cexp_of_re_mem_interior_integrableExpSet, euler_sineTerm_tprod, iteratedDeriv_odd_cos, ProbabilityTheory.integrable_pow_mul_cexp_of_re_mem_interior_integrableExpSet, ValueDistribution.characteristic_zero_mul_le, sin_sub, cpow_add, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add, log_eq_integral, cos_eq, Polynomial.Chebyshev.U_complex_cos, sin_add_sin, HasStrictFDerivAt.cpow, UpperHalfPlane.mul_eq_zero_iff, deriv_cpow_const, SchwartzMap.integral_fourierInv_mul_eq, ProbabilityTheory.complexMGF_id_gaussianReal, mul_conj, im_ofReal_mul, exp_rat_mul_pi_mul_I_pow_two_mul_den, qExpansion_mul, VectorFourier.fourierPowSMulRight_apply, HasDerivAt.cexp, Polynomial.Chebyshev.C_two_mul_complex_cos, cpow_int_mul', MeasureTheory.charFunDual_dirac, EisensteinSeries.G2_eq_tsum_cexp, Orientation.areaForm_map_complex, HadamardThreeLines.eventuallyle, sin_mul_I, GammaSeq_mul, qExpansion_mul_coeff_zero, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, sin_add_mul_I, ModularForm.cuspFunction_mul, EisensteinSeries.qExpansion_identity, iteratedDeriv_even_sin, GaussianFourier.integral_cexp_neg_sum_mul_add, HasStrictDerivAt.ccosh, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, arg_real_mul, rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar_val, LSeries.term_def₀, cosh_add, mul_inv_cancel, HurwitzZeta.hasSum_int_sinZeta, ZMod.toCircle_intCast, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, cos_sub_cos, ZMod.toCircle_natCast, inner_map_polarization, Real.deriv_fourierChar, cpow_eq_nhds', exp_eq_exp_re_mul_sin_add_cos, jacobiTheta_eq_tsum_nat, RCLike.norm_to_complex, Function.Periodic.qParam_left_inv_mod_period, cpow_def, taylorSeries_eq_on_eball', cpow_mul_int, hasDerivAt_Gamma_one_half, normSq_eq_conj_mul_self, ModularForm.slash_def, LSeries.delta_mul_eq_smul_delta, I_mul_im, Real.tendsto_integral_gaussian_smul, GammaIntegral_add_one, sin_sub_int_mul_two_pi, bernoulliFourierCoeff_recurrence, DirichletCharacter.Even.LFunction_neg_two_mul_nat_add_one, TemperedDistribution.smulLeftCLM_smulLeftCLM_apply, Gamma_mul_Gamma_add_half, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, sin_two_pi_sub, VectorFourier.fourierIntegral_probChar, PeriodPair.weierstrassPExceptSeries_hasSum, RCLike.to_complex_nonneg_iff, Orientation.kahler_rotation_right, MeasureTheory.charFun_dirac, HurwitzZeta.sinKernel_def, cot_eq_exp_ratio, LSeries.pow_mul_term_eq, exp_nat_mul, fourier_coe_apply', IsExpCmpFilter.isLittleO_cpow_exp, TemperedDistribution.lineDerivOp_fourierInv_eq, cos_sq, hasSum_taylorSeries_log, derivWithin_csinh, DirichletCharacter.LFunctionTrivChar_residue_one, norm_exp_mul_exp_add_exp_neg_le_of_abs_im_le, ProbabilityTheory.IsGaussian.charFun_eq, integral_gaussian_complex, polarCoord_symm_apply, Real.hasFDerivAt_fourierChar_neg_bilinear_right, summable_bernoulli_fourier, WeakFEPair.f_modif_aux1, ZMod.toCircle_apply, completedRiemannZeta_residue_one, GaussianFourier.integrable_cexp_neg_sum_mul_add, EisensteinSeries.linear_isTheta_right, DirichletCharacter.delta_mul, integral_exp_mul_complex_Iic, circleIntegral.integral_sub_center_inv, Polynomial.logMahlerMeasure_C_mul_X_add_C, isAlgebraic_tan_rat_mul_pi, iteratedDeriv_cexp_const_mul, completedRiemannZeta_one, SchwartzMap.fourier_fderivCLM_eq, DirichletCharacter.Odd.LFunction_neg_two_mul_nat_sub_one, Gammaℝ_div_Gammaℝ_one_sub, Real.iteratedDeriv_fourierIntegral, EisensteinSeries.summable_right_one_div_linear_sub_one_div_linear_succ, hasSum_sin', cos_eq_neg_one_iff, HasDerivWithinAt.cpow_const, Gamma_mul_Gamma_one_sub, DirichletCharacter.convolution_mul_moebius, EisensteinSeries.summable_linear_right_add_one_mul_linear_right, log_neg_I, log_mul_ofReal, betaIntegral_convergent_left, jacobiTheta₂_add_left', hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, cpow_int_mul, re_add_im, sin_eq_tsum', ValueDistribution.proximity_top_mul_le, ModularForm.SL_slash_apply, ZMod.LFunction_residue_one, PeriodPair.mul_ω₁_add_mul_ω₂_mem_lattice, norm_exp_ofReal_mul_I, Circle.coe_exp, EisensteinSeries.eisSummand_SL2_apply, exp_two_pi_mul_I, cpow_eq_nhds, tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero, fourier_add', ZMod.completedLFunction_one_sub_even, integral_boundary_rect_of_differentiableOn_real, sinh_sub, integral_mul_cpow_one_add_sq, IsExpCmpFilter.isLittleO_pow_mul_exp, HadamardThreeLines.scale_id_mem_verticalStrip_of_mem_verticalStrip, riemannZeta_residue_one, Orientation.oangle_map_complex, integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn, SlashInvariantForm.slash_action_eqn', tan_mul_I, Real.hasDerivAt_fourierIntegral, cos_nat_mul_two_pi_sub, rightAngleRotation, fourierCoeffOn_of_hasDerivAt, integrable_cexp_quadratic, image_exp_Ioc_eq_sphere, HurwitzZeta.completedHurwitzZetaEven_residue_zero, UpperHalfPlane.denom_cocycle', ArithmeticFunction.iteratedDeriv_LSeries_alternating, two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, Real.tsum_eq_tsum_fourierIntegral, exp_eq_one_iff, cos_nat_mul_two_pi_add_pi, SchwartzMap.tsum_eq_tsum_fourierIntegral, inv_Gammaℝ_one_sub, Orientation.kahler_rightAngleRotation_right, cpow_mul_ofNat, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, exp_mul_I_antiperiodic, hasSum_arctan, fourier_add, cpow_def_of_ne_zero, exp_mul_I, ValueDistribution.characteristic_top_mul_eventually_le, isAlgebraic_cos_rat_mul_pi, HurwitzZeta.hasSum_int_completedCosZeta, TorusIntegrable.function_integrable, riemannZeta_neg_nat_eq_bernoulli, DirichletCharacter.LFunction_isBigO_horizontal, taylorSeries_eq_of_entire', HasFDerivAt.cpow, isometryOfOrthonormal_symm_apply, arg_mul_cos_add_sin_mul_I_sub, PeriodPair.summable_weierstrassPExceptSummand, integrable_mul_cexp_neg_mul_sq, NumberField.InfiniteAdeleRing.ringEquiv_mixedSpace_apply, hasDerivAt_log_sub_logTaylor, GammaIntegral_convergent, HurwitzZeta.sinZeta_eq, cos_two_pi_sub, ProbabilityTheory.IsGaussian.charFunDual_eq', enorm_exp_ofReal_mul_I, riemannZeta_two_mul_nat, Real.probChar_apply, ModularForm.logDeriv_qParam, tendsto_euler_sin_prod', ProbabilityTheory.charFun_gaussianReal, HasDerivAt.const_cpow, SlashInvariantForm.coe_mul, arg_mul_coe_angle, sin_sub_sin, hasSum_cos, logTaylor_succ, HurwitzZeta.hurwitzZetaOdd_one_sub, HurwitzZeta.hasSum_int_completedSinZeta, eqOn_iteratedDerivWithin_cotTerm_integerComplement, ProbabilityTheory.IndepFun.charFun_map_add_eq_mul, Real.fourierInv_eq', sinh_mul_I, Real.fourierIntegral_deriv, norm_exp_I_mul_ofReal, DirichletCharacter.LSeries_eulerProduct, integrable_cexp_quadratic', sin_add_nat_mul_two_pi, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, cuspFunction_mul_zero, norm_mul, exp_mul_I_periodic, log_inv_eq_integral, ZMod.LFunction_dft, cfcHom_eq_of_isStarNormal, betaIntegral_eq_Gamma_mul_div, Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay, cos_eq_cos_iff, integral_cpow_mul_exp_neg_mul_Ioi, range_exp_mul_I, MeasureTheory.charFun_map_const_add, integral_cos_mul_complex, IsExpCmpFilter.isLittleO_cpow_mul_exp, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, ZMod.completedLFunction_modOne_eq, summable_pow_mul_cexp, HurwitzZeta.sinZeta_one_sub, BoundedContinuousFunction.char_add_eq_mul, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, approx_Gamma_integral_tendsto_Gamma_integral, tendsto_tsum_powerSeries_nhdsWithin_stolzSet, isSquare, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq, MeasureTheory.charFun_prod, summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp, inv_Gammaℝ_two_sub, Real.fderiv_fourierChar_neg_bilinear_left_apply, Real.fourier_eq', MeasureTheory.charFunDual_eq_prod_iff', EisensteinSeries.summand_bound, cosh_mul_I, cos_int_mul_two_pi, HadamardThreeLines.scale_id_mem_verticalClosedStrip_of_mem_verticalClosedStrip, integrable_cexp_neg_mul_sq, cos_two_pi, cos_add_int_mul_two_pi, mul_I_im, continuousFunctionalCalculus_map_id, fourierCoeffOn_of_hasDeriv_right, normSq_ofReal_sub_I_mul_sqrt_one_sub, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, exp_int_mul, ModularForm.mul_coe, exp_int_mul_two_pi_mul_I, arg_cos_add_sin_mul_I_eq_toIocMod, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, LSeries_eq_mul_integral', log_exp_exists, horizontalSegment_eq, ofReal_mul, fourier_gaussian_pi, continuousAt_gaussian_integral, Orientation.rightAngleRotation_map_complex, Real.deriv_fourier, taylorSeries_eq_on_emetric_ball', UnitDisc.coe_circle_smul, NumberField.mixedEmbedding.logMap_mul, EulerSine.integral_cos_mul_cos_pow_even, circleIntegral.integral_const_mul, LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div, LSeries_convolution, ValueDistribution.characteristic_mul_top_eventuallyLE, HasStrictDerivAt.ccos, ModularGroup.smul_eq_lcRow0_add, Gammaℝ_mul_Gammaℝ_add_one, fourierIntegral_gaussian_pi', tendsto_tsum_powerSeries_nhdsWithin_lt, tan_two_mul, integral_gaussian_complex_Ioi, AnalyticOnNhd.eq_re_add_const_mul_I_of_re_eq_const, MeasureTheory.charFun_map_add_const, cpow_ofReal, rotation, two_sin, RCLike.complexLinearIsometryEquiv_apply, differentiableAt_iteratedDerivWithin_cexp, pi_mul_cot_pi_q_exp, BoundedContinuousFunction.probCharDual_apply, SchwartzMap.tsum_eq_tsum_fourier, hasDerivAt_Gamma_nat, HasFDerivAt.const_cpow, cos_sub_int_mul_two_pi, tan_eq_one_sub_tan_half_sq_div_one_add_tan_half_sq, EisensteinSeries.linear_inv_isBigO_right, ModularForm.slash_apply, integrableOn_exp_mul_complex_Ioi, angle_exp_exp, ProbabilityTheory.indepFun_iff_charFunDual_prod, two_cos, HasDerivAt.cpow_const, re_ofReal_mul, lim_eq_lim_im_add_lim_re, MeasureTheory.charFunDual_conv, rectangle_eq_convexHull, HurwitzZeta.hurwitzZetaEven_residue_one, UpperHalfPlane.petersson_slash, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
instNNRatCast 📖	CompOp	5 math math: im_nnratCast, norm_nnratCast, ofReal_nnratCast, nnnorm_nnratCast, re_nnratCast
instNatCast 📖	CompOp	554 math math: ArithmeticFunction.vonMangoldt.residueClass_eq, HurwitzZeta.hasSum_int_hurwitzZetaOdd, Real.hasDerivAt_fourier, Polynomial.Chebyshev.S_two_mul_complex_cos, hasSum_mellin_pi_mul_sq', cos_eq_tsum, EulerSine.integral_cos_mul_cos_pow, ofNat_log, log_I, ProbabilityTheory.isGaussian_iff_gaussian_charFun, completedZeta_eq_tsum_of_one_lt_re, PeriodPair.hasFPowerSeriesAt_weierstrassPExcept, Real.hasFDerivAt_fourierChar_neg_bilinear_left, cos_three_mul, EisensteinSeries.hasSum_e2Summand_symmetricIcc, Gammaℝ_def, sin_pi_div_two_sub, HurwitzZeta.hasSum_int_cosKernel₀, exp_eq_exp_iff_exists_int, HurwitzZeta.hasSum_int_hurwitzZetaEven, re_eq_add_conj, normSq_natCast, tan_nat_mul_pi_sub, DirichletCharacter.LSeries_eulerProduct_tprod, PeriodPair.iteratedDeriv_weierstrassPExcept_self, PeriodPair.derivWeierstrassPExcept_sub, tsum_eisSummand_eq_riemannZeta_mul_eisensteinSeries, ProbabilityTheory.hasGaussianLaw_iff_charFunDual_map_eq, Gamma_neg_nat_eq_zero, HurwitzZeta.cosZeta_two_mul_nat', Gamma_nat_eq_factorial, summableLocallyUniformlyOn_iteratedDerivWithin_cexp, cos_eq_one_iff, PeriodPair.ω₁_div_two_notMem_lattice, ModularForm.eta_q_eq_pow, HurwitzZeta.sinZeta_neg_two_mul_nat_sub_one, ZMod.LFunction_one_sub, natCast_re, hasSum_taylorSeries_of_entire, tsum_exp_neg_mul_int_sq, Real.fderiv_fourierChar_neg_bilinear_right_apply, ZMod.invDFT_apply, SchwartzMap.lineDerivOp_fourier_eq, GaussianFourier.integral_cexp_neg_mul_sq_norm, circleIntegral.integral_sub_inv_of_mem_ball, hasSum_one_div_nat_pow_mul_fourier, two_mul_riemannZeta_eq_tsum_int_inv_pow_of_even, cos_eq_zero_iff, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, cos_add_nat_mul_two_pi, TemperedDistribution.lineDerivOp_fourier_eq, HurwitzZeta.cosZeta_two_mul_nat, sin_eq_neg_one_iff, riemannZeta_two, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, HurwitzZeta.cosZeta_neg_two_mul_nat_add_one, fourierIntegral_gaussian_innerProductSpace', fourier_gaussian_pi', iteratedDerivWithin_tsum_cexp_eq, PeriodPair.iteratedDeriv_derivWeierstrassPExcept_self, jacobiTheta₂_functional_equation, ProbabilityTheory.hasGaussianLaw_iff_charFun_map_eq, hasSum_taylorSeries_on_ball, cosh_three_mul, GaussianFourier.integral_cexp_neg_mul_sq_norm_add, sum_cauchyPowerSeries_eq_integral, tan_pi_div_two_sub, taylorSeries_eq_of_entire, not_differentiableAt_Gamma_neg_nat, HurwitzZeta.jacobiTheta₂'_functional_equation', cos_pi_div_two, hasSum_nat_jacobiTheta, Real.fourier_deriv, DirichletCharacter.norm_LSeries_product_ge_one, PeriodPair.hasSum_derivWeierstrassPExcept, ArithmeticFunction.LSeries_zeta_mul_Lseries_moebius, digamma_one_half, ZMod.stdAddChar_coe, DifferentiableOn.circleIntegral_sub_inv_smul, ArithmeticFunction.abscissaOfAbsConv_zeta, ZMod.completedLFunction_def_even, sin_nat_mul_pi, ModularForm.tsum_logDeriv_eta_q, integral_circleTransform, DirichletCharacter.LFunction_changeLevel, HurwitzZeta.oddKernel_def', cos_eq_two_mul_tan_half_div_one_sub_tan_half_sq, fourier_gaussian_innerProductSpace, tan_nat_mul_pi, DirichletCharacter.IsPrimitive.completedLFunction_one_sub, isPrimitiveRoot_exp_rat, VectorFourier.fourierPowSMulRight_eq_comp, EisensteinSeries.E2_slash_action, mem_rootsOfUnity, sin_int_mul_two_pi_sub, cos_nat_mul_two_pi_sub_pi, sin_bound, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, two_sinh, HurwitzZeta.hasSum_nat_sinZeta, taylorSeries_eq_on_ball', fourierCoeff_bernoulli_eq, circleMap_eq_circleMap_iff, ZMod.LFunction_def_odd, EulerSine.integral_cos_mul_cos_pow_aux, SchwartzMap.fourier_lineDerivOp_eq, TemperedDistribution.fourierInv_lineDerivOp_eq, AddChar.sum_apply_eq_ite, cpow_ofNat_mul', cos_eq_iff_quadratic, LSeries.term_def, riemannZeta_one_sub, PeriodPair.ω₂_div_two_notMem_lattice, PeriodPair.hasSum_derivWeierstrassP, sinh_three_mul, natCast_im, PeriodPair.hasSumLocallyUniformly_derivWeierstrassP, cos_nat_mul_two_pi, natCast_mul_natCast_cpow, PeriodPair.hasFPowerSeriesAt_derivWeierstrassPExcept, cauchyPowerSeries_apply, HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re, sin_sub_nat_mul_two_pi, DiffContOnCl.circleIntegral_sub_inv_smul, HurwitzZeta.hurwitzZetaEven_apply_zero, sin_nat_mul_two_pi_sub, sin_two_mul, HurwitzZeta.expZeta_one_sub, HurwitzZeta.cosZeta_apply_zero, norm_natCast, ArithmeticFunction.vonMangoldt.residueClass_apply, fourierIntegral_gaussian_pi, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, Real.circleAverage_eq_circleIntegral, cos_eq_tsum', mellinInv_eq_fourierIntegralInv, cot_series_rep', natCast_log, ModularForm.coe_natCast, hasDerivAt_Gammaℝ_one, riemannZeta_four, cos_two_mul, HurwitzZeta.hasSum_nat_completedCosZeta, tendsto_sub_mul_tsum_nat_cpow, cos_sub_two_pi, sin_sub_two_pi, sin_sub_pi_div_two, DirichletCharacter.LSeries_changeLevel, cos_add_sin_mul_I_pow, mellinInv_eq_fourierInv, fourier_gaussian_innerProductSpace', HurwitzZeta.hasSum_nat_hurwitzZetaEven, LSeries.term_of_ne_zero, HurwitzZeta.hasSum_int_cosKernel, ZMod.completedLFunction_def_odd, Real.tendsto_integral_gaussian_smul', HurwitzZeta.hurwitzZeta_neg_nat, EisensteinSeries.tendsto_zero_inv_linear_sub, exp_neg_pi_div_two_mul_I, ZetaAsymptotics.tendsto_Gamma_term_aux, exp_nsmul', inner_map_polarization', EisensteinSeries.D2_S, Polynomial.int_cyclotomic_rw, ZMod.LFunction_neg_two_mul_nat_sub_one, tan_int_mul_pi_div_two, EisensteinSeries.G2_S_transform, tendsto_one_add_div_pow_exp, natCast_mem_slitPlane, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, riemannZeta_eulerProduct_exp_log, cos_int_mul_two_pi_sub, ArithmeticFunction.LSeries_zeta_eq, Real.fourierIntegral_iteratedDeriv, ProbabilityTheory.IsGaussian.charFun_eq', HurwitzZeta.cosZeta_one_sub, EulerSine.sin_pi_mul_eq, integral_mul_cexp_neg_mul_sq, jacobiTheta_S_smul, zeta_eq_tsum_one_div_nat_cpow, isPrimitiveRoot_iff, exp_periodic, exp_nat_mul_two_pi_mul_I, log_exp_eq_sub_toIocDiv, norm_exp_sub_sum_le_norm_mul_exp, deriv_Gamma_nat, ProbabilityTheory.complexMGF_gaussianReal, isPrimitiveRoot_exp_of_isCoprime, Gammaℝ_eq_zero_iff, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one, Polynomial.Chebyshev.C_two_mul_complex_cosh, cpow_nat_mul, integral_exp_mul_I_eq_sinc, HurwitzZeta.cosZeta_eq, sin_two_pi, UpperHalfPlane.norm_exp_two_pi_I_lt_one, HurwitzZeta.hasSum_int_cosZeta, isCauSeq_norm_exp, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, tendsto_norm_tan_atTop, ProbabilityTheory.IsGaussian.charFunDual_eq_of_forall_strongDual_eq_zero, hasDerivAt_Gammaℂ_one, hasSum_sin, riemannZeta_neg_two_mul_nat_add_one, sin_add_two_pi, EisensteinSeries.hasSum_e2Summand_symmetricIco, ModularForm.eta_q_eq_cexp, ArithmeticFunction.convolution_vonMangoldt_const_one, circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const, EisensteinSeries.q_expansion_riemannZeta, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, fourier_coe_apply, Real.deriv_fourierIntegral, integral_exp_mul_I_eq_sin, eisSummand_of_gammaSet_eq_divIntMap, ArithmeticFunction.convolution_vonMangoldt_zeta, jacobiTheta₂'_add_right, HurwitzZeta.hasSum_nat_cosZeta, DirichletCharacter.LFunctionTrivChar_eq_mul_riemannZeta, hasSum_cauchyPowerSeries_integral, pow_cpow_ofNat_inv, sin_eq_sin_iff, ProbabilityTheory.isGaussian_iff_charFun_eq, fourierIntegral_eq_half_sub_half_period_translate, has_antideriv_at_fourier_neg, HurwitzZeta.hasSum_expZeta_of_one_lt_re, sin_three_mul, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, EisensteinSeries.e2Summand_zero_eq_two_riemannZeta_two, one_div_sub_sq_hasFPowerSeriesOnBall_zero, DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow, ZMod.invDFT_def', two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, PeriodPair.derivWeierstrassP_sq, cos_sub_nat_mul_two_pi, cos_int_mul_two_pi_add_pi, eqOn_iteratedDeriv_cotTerm, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, hasDerivAt_fourier_neg, SchwartzMap.fourierInv_lineDerivOp_eq, cos_two_mul', tendsto_euler_sin_prod, MeasureTheory.integral_charFun_Icc, ZMod.completedLFunction_eq, EulerSine.integral_sin_mul_sin_mul_cos_pow_eq, ProbabilityTheory.IsGaussian.charFunDual_eq, tsum_exp_neg_quadratic, ModularFormClass.qExpansion_coeff, riemannZeta_zero, isAddQuotientCoveringMap_exp, cot_pi_eq_exp_ratio, cosh_eq_tsum, isIntegral_two_mul_sin_rat_mul_pi, HurwitzZeta.hurwitzZetaEven_one_sub, HurwitzZeta.hasSum_int_sinKernel, ZMod.invDFT_apply', SlashInvariantForm.coe_natCast, two_cosh, cot_series_rep, HurwitzZeta.LSeriesHasSum_exp, DirichletCharacter.convolution_twist_vonMangoldt, bernoulliFourierCoeff_eq, sinh_two_mul, cosh_two_mul, isPrimitiveRoot_exp_of_coprime, tan_sub_nat_mul_pi, inv_natCast_cpow_ofReal_pos, ModularForm.logDeriv_eta_eq_E2, TemperedDistribution.fourier_lineDerivOp_eq, sin_eq_one_iff, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, Real.fourier_iteratedDeriv, cos_periodic, cpow_mul_nat, exp_pi_div_two_mul_I, VectorFourier.fourierSMulRight_apply, integral_cexp_quadratic, GammaSeq_add_one_left, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, HurwitzZeta.sinZeta_two_mul_nat_add_one', Gammaℂ_def, cos_bound, cpow_natCast, cpow_inv_two_im_eq_sqrt, cpow_inv_two_re, NumberField.mixedEmbedding.det_matrixToStdBasis, hasDerivAt_fourier, ZMod.LFunction_def_even, one_div_sub_pow_hasFPowerSeriesOnBall_zero, isPrimitiveRoot_exp, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, HurwitzZeta.hasSum_nat_completedSinZeta, hasSum_taylorSeries_on_emetric_ball, norm_natCast_cpow_of_pos, hasFPowerSeriesOn_cauchy_integral, circleIntegral_div_sub_of_differentiable_on_off_countable, contDiffOn_tsum_cexp, Gammaℂ_add_one, cos_add_pi_div_two, fourierIntegral_gaussian_innerProductSpace, ProbabilityTheory.HasGaussianLaw.charFun_map_eq, EisensteinSeries.qExpansion_identity_pnat, Real.iteratedDeriv_fourier, zeta_eq_tsum_one_div_nat_add_one_cpow, norm_prime_cpow_le_one_half, hasSum_cos', cos_add_cos, ZMod.dft_dft, jacobiTheta₂'_add_left', jacobiTheta₂'_functional_equation, riemannZeta_one, exp_sub_sum_range_succ_isLittleO_pow, pow_cpow_nat_inv, natCast_cpow_natCast_mul, sin_pi_div_two, sin_periodic, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, GammaSeq_eq_betaIntegral_of_re_pos, cos_sub_pi_div_two, EisensteinSeries.tendsto_zero_inv_linear, norm_natCast_cpow_pos_of_pos, LSeries.term_of_ne_zero', exp_sub_sum_range_isBigO_pow, riemannZeta_eulerProduct, DirichletCharacter.norm_LFunction_product_ge_one, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, cos_int_mul_two_pi_sub_pi, HurwitzZeta.differentiableAt_update_of_residue, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, cos_add_two_pi, exp_bound, taylorSeries_eq_on_eball, Real.hasDerivAt_fourierChar, ZMod.LFunction_neg_two_mul_nat_add_one, DirichletCharacter.LSeries_eulerProduct_hasProd, PeriodPair.deriv_derivWeierstrassPExcept_self, EisensteinSeries.q_expansion_bernoulli, cotTerm_identity, HurwitzZeta.sinZeta_two_mul_nat_add_one, Polynomial.Chebyshev.S_two_mul_complex_cosh, ArithmeticFunction.LSeries_zeta_eq_riemannZeta, nnnorm_natCast, completedRiemannZeta₀_one, fourierCoeffOn_of_hasDerivAt_Ioo, HurwitzZeta.hurwitzZetaOdd_eq, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, SchwartzMap.lineDerivOp_fourierInv_eq, DirichletCharacter.LSeries_eulerProduct_exp_log, ZMod.completedLFunction_one_sub_odd, DirichletCharacter.Even.LFunction_neg_two_mul_nat, Gammaℝ_residue_zero, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, EulerSine.antideriv_cos_comp_const_mul, GaussianFourier.integral_cexp_neg_mul_sum_add, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat_sub_one, ModularFormClass.qExpansion_coeff_eq_circleIntegral, ProbabilityTheory.isGaussian_iff_charFunDual_eq, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, sin_eq_two_mul_tan_half_div_one_add_tan_half_sq, fourierIntegral_gaussian, LSeries.tendsto_cpow_mul_atTop, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, sinh_eq_tsum, sin_add_int_mul_two_pi, HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc, im_eq_sub_conj, EulerSine.antideriv_sin_comp_const_mul, div_natCast_im, ofReal_natCast, differentiableAt_Gamma_nat_add_one, HurwitzZeta.hurwitzZeta_one_sub, ArithmeticFunction.LSeriesSummable_zeta_iff, DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, taylorSeries_eq_on_emetric_ball, not_continuousAt_Gamma_neg_nat, isIntegral_two_mul_cos_rat_mul_pi, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun, sin_eq_tsum, Gammaℝ_one_sub_mul_Gammaℝ_one_add, tan_add_nat_mul_pi, cpow_nat_mul', SchwartzMap.fderivCLM_fourier_eq, tan_eq_zero_iff, Gammaℝ_add_two, hasSum_taylorSeries_neg_log, taylorSeries_eq_on_ball, exp_bound', sin_add_sin, cpow_nat_inv_pow, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, ZMod.LFunction_apply_zero_of_even, ProbabilityTheory.complexMGF_id_gaussianReal, summable_one_div_nat_cpow, VectorFourier.fourierPowSMulRight_apply, Polynomial.Chebyshev.C_two_mul_complex_cos, conj_natCast, EisensteinSeries.G2_eq_tsum_cexp, GammaSeq_mul, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, EisensteinSeries.qExpansion_identity, GaussianFourier.integral_cexp_neg_sum_mul_add, ArithmeticFunction.LSeriesHasSum_zeta, norm_cpow_inv_nat, cos_pi_div_two_sub, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar_val, LSeries.term_def₀, HurwitzZeta.hasSum_int_sinZeta, ZMod.toCircle_intCast, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, cos_sub_cos, ArithmeticFunction.LSeries_zeta_eulerProduct_exp_log, ZMod.toCircle_natCast, inner_map_polarization, Real.deriv_fourierChar, jacobiTheta_eq_tsum_nat, taylorSeries_eq_on_eball', hasDerivAt_Gamma_one_half, ZMod.invDFT_def, Real.tendsto_integral_gaussian_smul, sin_sub_int_mul_two_pi, bernoulliFourierCoeff_recurrence, DirichletCharacter.Even.LFunction_neg_two_mul_nat_add_one, Gamma_mul_Gamma_add_half, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, cpow_inv_two_im_eq_neg_sqrt, sin_two_pi_sub, HurwitzZeta.sinKernel_def, cot_eq_exp_ratio, LSeries.pow_mul_term_eq, norm_natCast_cpow_le_norm_natCast_cpow_iff, norm_natCast_cpow_le_norm_natCast_cpow_of_pos, exp_nat_mul, fourier_coe_apply', TemperedDistribution.lineDerivOp_fourierInv_eq, cos_sq, hasSum_taylorSeries_log, ProbabilityTheory.IsGaussian.charFunDual_eq_of_integral_eq_zero, DirichletCharacter.LFunctionTrivChar_residue_one, ProbabilityTheory.IsGaussian.charFun_eq, integral_gaussian_complex, Real.hasFDerivAt_fourierChar_neg_bilinear_right, summable_bernoulli_fourier, ZMod.toCircle_apply, hasSum_sinh, circleIntegral.integral_sub_center_inv, norm_exp_sub_sum_le_exp_norm_sub_sum, norm_log_natCast_le_rpow_div, completedRiemannZeta_one, SchwartzMap.fourier_fderivCLM_eq, DirichletCharacter.Odd.LFunction_neg_two_mul_nat_sub_one, Gammaℝ_div_Gammaℝ_one_sub, Real.iteratedDeriv_fourierIntegral, hasSum_sin', cos_eq_neg_one_iff, log_neg_I, jacobiTheta₂_add_left', hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, riemannZeta_neg_nat_eq_bernoulli', sin_eq_tsum', betaIntegral_eval_nat_add_one_right, ZMod.LFunction_residue_one, riemannZeta_eulerProduct_tprod, exp_two_pi_mul_I, ZMod.completedLFunction_one_sub_even, integral_mul_cpow_one_add_sq, div_natCast, Real.hasDerivAt_fourierIntegral, cos_nat_mul_two_pi_sub, natCast_arg, fourierCoeffOn_of_hasDerivAt, two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, jacobiTheta₂_add_right, exp_eq_one_iff, cos_nat_mul_two_pi_add_pi, inv_Gammaℝ_one_sub, norm_two, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, hasSum_arctan, HurwitzZeta.hasSum_int_completedCosZeta, hasSum_cosh, riemannZeta_neg_nat_eq_bernoulli, taylorSeries_eq_of_entire', Gamma_ofNat_eq_factorial, HurwitzZeta.sinZeta_eq, cos_two_pi_sub, ProbabilityTheory.IsGaussian.charFunDual_eq', riemannZeta_two_mul_nat, abs_cpow_inv_two_im, UpperHalfPlane.im_pnat_div_pos, ModularForm.logDeriv_qParam, ProbabilityTheory.charFun_gaussianReal, sin_sub_sin, hasSum_cos, logTaylor_succ, HurwitzZeta.hurwitzZetaOdd_one_sub, HurwitzZeta.hasSum_int_completedSinZeta, eqOn_iteratedDerivWithin_cotTerm_integerComplement, zeta_nat_eq_tsum_of_gt_one, HurwitzZeta.hasSum_nat_hurwitzZetaOdd_of_mem_Icc, Real.fourierIntegral_deriv, riemannZeta_eulerProduct_hasProd, isCauSeq_exp, DirichletCharacter.LSeries_eulerProduct, sin_add_nat_mul_two_pi, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, exp_mul_I_periodic, Gamma_eq_zero_iff, HurwitzZeta.hasSum_int_completedHurwitzZetaEven, cos_eq_cos_iff, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, summable_pow_mul_cexp, HurwitzZeta.sinZeta_one_sub, cpow_two, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq, summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp, inv_Gammaℝ_two_sub, Real.fderiv_fourierChar_neg_bilinear_left_apply, Gamma_one_half_eq, cos_int_mul_two_pi, sq_cpow_two_inv, cos_two_pi, cos_add_int_mul_two_pi, fourierCoeffOn_of_hasDeriv_right, PeriodPair.derivWeierstrassPExcept_def, exp_int_mul_two_pi_mul_I, div_natCast_re, log_exp_exists, fourier_gaussian_pi, Real.deriv_fourier, taylorSeries_eq_on_emetric_ball', EulerSine.integral_cos_mul_cos_pow_even, norm_natCast_cpow_of_re_ne_zero, fourierIntegral_gaussian_pi', tan_two_mul, HurwitzZeta.hurwitzZetaEven_eq, integral_gaussian_complex_Ioi, hasSum_mellin_pi_mul_sq, jacobiTheta_two_add, two_sin, differentiableAt_iteratedDerivWithin_cexp, pi_mul_cot_pi_q_exp, hasDerivAt_Gamma_nat, cos_sub_int_mul_two_pi, tan_eq_one_sub_tan_half_sq_div_one_add_tan_half_sq, sin_add_pi_div_two, two_cos, hasSum_taylorSeries_on_eball, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
instNeg 📖	CompOp	378 math math: HurwitzZeta.hasSum_int_hurwitzZetaOdd, HurwitzZeta.jacobiTheta₂''_neg_left, cos_sub_pi, Real.hasDerivAt_fourier, hasSum_mellin_pi_mul_sq', cos_eq_tsum, norm_log_one_sub_inv_add_logTaylor_neg_le, HasDerivAt.ccos, jacobiTheta₂_conj, log_neg_one, sin_eq_zero_iff_cos_eq, completedZeta_eq_tsum_of_one_lt_re, HasFDerivWithinAt.ccos, GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I, Real.hasFDerivAt_fourierChar_neg_bilinear_left, Gammaℝ_def, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, subset_slitPlane_iff_of_subset_sphere, cos_sub_sin_I, tan_nat_mul_pi_sub, DirichletCharacter.LSeries_eulerProduct_tprod, isLittleO_exp_neg_mul_sq_cocompact, Gamma_neg_nat_eq_zero, HurwitzZeta.cosZeta_two_mul_nat', inv_I, HurwitzZeta.sinZeta_neg_two_mul_nat_sub_one, Orientation.kahler_comp_rightAngleRotation', ZMod.LFunction_one_sub, arg_neg_I, DirichletCharacter.continuousOn_neg_logDeriv_LFunction_of_nontriv, tsum_exp_neg_mul_int_sq, Real.fderiv_fourierChar_neg_bilinear_right_apply, SchwartzMap.lineDerivOp_fourier_eq, tan_pi_sub, GaussianFourier.integral_cexp_neg_mul_sq_norm, hasSum_one_div_nat_pow_mul_fourier, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, cos_antiperiodic, TemperedDistribution.lineDerivOp_fourier_eq, HurwitzZeta.cosZeta_two_mul_nat, sin_eq_neg_one_iff, HurwitzZeta.cosZeta_neg_two_mul_nat_add_one, fourierIntegral_gaussian_innerProductSpace', fourier_gaussian_pi', jacobiTheta₂_functional_equation, GaussianFourier.integral_cexp_neg_mul_sq_norm_add, norm_cexp_neg_mul_sq, UpperHalfPlane.num_neg, DirichletCharacter.LSeries_twist_vonMangoldt_eq, HurwitzZeta.evenKernel_def, cpow_neg_one, deriv_cos, not_differentiableAt_Gamma_neg_nat, HurwitzZeta.jacobiTheta₂'_functional_equation', PeriodPair.hasSum_derivWeierstrassPExcept, digamma_one_half, UnitDisc.coe_neg, ZMod.completedLFunction_def_even, jacobiTheta₂'_conj, ModularForm.tsum_logDeriv_eta_q, DirichletCharacter.LFunction_changeLevel, HurwitzZeta.oddKernel_def', fourier_gaussian_innerProductSpace, VectorFourier.fourierPowSMulRight_eq_comp, sin_int_mul_two_pi_sub, TemperedDistribution.smulLeftCLM_neg, cos_nat_mul_two_pi_sub_pi, NumberField.mixedEmbedding.convexBodySum_neg_mem, LSeries_iteratedDeriv, ModularForm.coe_neg, two_sinh, exp_antiperiodic, fourierCoeff_bernoulli_eq, exp_add_pi_mul_I, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace, log_inv_eq_ite, TemperedDistribution.fourierInv_lineDerivOp_eq, arg_of_re_neg_of_im_nonneg, riemannZeta_one_sub, PeriodPair.derivWeierstrassP_neg, PeriodPair.hasSum_derivWeierstrassP, integral_exp_mul_complex_Ioi, LSeries_deriv_eqOn, PeriodPair.hasSumLocallyUniformly_derivWeierstrassP, HurwitzZeta.completedSinZeta_neg, HurwitzZeta.hurwitzZetaEven_apply_zero, arg_neg_eq_arg_sub_pi_of_im_pos, ofReal_cpow_of_nonpos, sin_nat_mul_two_pi_sub, jacobiTheta₂_neg_left, HurwitzZeta.expZeta_one_sub, HurwitzZeta.cosZeta_apply_zero, Unitary.norm_sub_one_lt_two_iff, PeriodPair.weierstrassPExcept_neg, log_inv, fourierIntegral_gaussian_pi, LSeries_neg, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, mellinInv_eq_fourierIntegralInv, cos_pi_sub, hasDerivAt_Gammaℝ_one, ofReal_neg, LSeries_deriv, Function.Periodic.cuspFunction_neg, sin_sub_pi_div_two, DirichletCharacter.LSeries_changeLevel, tan_neg, mellinInv_eq_fourierInv, fourier_gaussian_innerProductSpace', sqrt_neg_of_nonneg, tanh_neg, Real.tendsto_integral_gaussian_smul', HurwitzZeta.hurwitzZeta_neg_nat, exp_neg_pi_div_two_mul_I, GaussianFourier.integrable_cexp_neg_mul_sum_add, ZetaAsymptotics.tendsto_Gamma_term_aux, LSeries_eq_mul_integral_of_nonneg, EisensteinSeries.G2_S_transform, riemannZeta_eulerProduct_exp_log, iteratedDeriv_odd_sin, HurwitzZeta.cosZeta_one_sub, arg_neg_eq_arg_add_pi_iff, integral_mul_cexp_neg_mul_sq, jacobiTheta_S_smul, cuspFunction_neg, fderivWithin_ccos, deriv_Gamma_nat, liftAux_neg_I, cpow_neg, Gammaℝ_eq_zero_iff, CuspForm.coe_neg, HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one, ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div, arg_of_re_neg_of_im_neg, UpperHalfPlane.modular_S_smul, cos_add_pi, ProbabilityTheory.IsGaussian.charFunDual_eq_of_forall_strongDual_eq_zero, HasStrictFDerivAt.ccos, hasDerivAt_Gammaℂ_one, hasSum_sin, riemannZeta_neg_two_mul_nat_add_one, EisensteinSeries.D2_inv, EisensteinSeries.q_expansion_riemannZeta, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, Real.deriv_fourierIntegral, sinh_neg, DirichletCharacter.LFunctionTrivChar_eq_mul_riemannZeta, HasFDerivAt.ccos, has_antideriv_at_fourier_neg, exp_neg, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow, hasDerivAt_cos, ModularForm.logDeriv_one_sub_mul_cexp_comp, jacobiTheta₂'_neg_left, arg_neg_eq_arg_add_pi_of_im_neg, cos_int_mul_two_pi_add_pi, eqOn_iteratedDeriv_cotTerm, NumberField.mixedEmbedding.convexBodySumFun_neg, hasDerivAt_fourier_neg, SchwartzMap.fourierInv_lineDerivOp_eq, ZMod.completedLFunction_eq, LSeriesSummable.neg_iff, tsum_exp_neg_quadratic, riemannZeta_zero, skewAdjoint.negISMul_apply_coe, ModularForm.neg_apply, HurwitzZeta.hurwitzZetaEven_one_sub, hasSum_mellin_pi_mul, HurwitzZeta.hasSum_int_sinKernel, imaginaryPart_eq_neg_I_smul_skewAdjointPart, two_cosh, sin_antiperiodic, HurwitzZeta.completedCosZeta_residue_zero, bernoulliFourierCoeff_eq, hasDerivAt_logTaylor, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, VectorFourier.fourierSMulRight_apply, integral_cexp_quadratic, UpperHalfPlane.J_smul, HurwitzZeta.sinZeta_two_mul_nat_add_one', imaginaryPart_apply_coe, Gammaℂ_def, LSeries.term_neg_apply, iteratedDeriv_even_cos, UpperHalfPlane.denom_neg, CuspForm.neg_apply, HurwitzZeta.completedHurwitzZetaOdd_neg, fderiv_ccos, ZMod.LFunction_def_even, ModularForm.logDeriv_one_sub_cexp, isIntegral_exp_neg_rat_mul_pi_mul_I, LSeries_eq_mul_integral, cos_add_pi_div_two, hasStrictDerivAt_cos, fourierIntegral_gaussian_innerProductSpace, iteratedDeriv_add_one_cos, ArithmeticFunction.LSeries_vonMangoldt_eq, EisensteinSeries.qExpansion_identity_pnat, sqrt_neg_one, Real.iteratedDeriv_fourier, NumberField.mixedEmbedding.convexBodyLT_neg_mem, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', norm_prime_cpow_le_one_half, jacobiTheta₂'_add_left', jacobiTheta₂'_functional_equation, mellin_comp_inv, derivWithin_ccos, LSeries.term_neg, UpperHalfPlane.coe_J_smul, hasSum_arctan_aux, arg_neg_one, tan_int_mul_pi_sub, riemannZeta_eulerProduct, hasSum_mellin_pi_mul₀, I_pow_three, Orientation.kahler_rightAngleRotation_left, SlashInvariantForm.coe_neg, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, NumberField.mixedEmbedding.normAtPlace_neg, cos_int_mul_two_pi_sub_pi, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, ZMod.LFunction_neg_two_mul_nat_add_one, LindemannWeierstrass.integral_exp_mul_eval, LSeries.iteratedDeriv_alternating, DirichletCharacter.LSeries_eulerProduct_hasProd, Real.tendsto_integral_cexp_sq_smul, deriv_ccos, sqrt_neg_I, HurwitzZeta.sinZeta_two_mul_nat_add_one, sqrt_eq_real_add_ite, HasDerivWithinAt.ccos, fourierCoeffOn_of_hasDerivAt_Ioo, cos_neg, DirichletCharacter.LSeries_eulerProduct_exp_log, DirichletCharacter.Even.LFunction_neg_two_mul_nat, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, GaussianFourier.integral_cexp_neg_mul_sum_add, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat_sub_one, neg_re, fourierIntegral_gaussian, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, EulerSine.antideriv_sin_comp_const_mul, GaussianInt.toComplex_neg, HurwitzZeta.hurwitzZeta_one_sub, deriv_cos', integral_gaussian_sq_complex, qExpansion_neg, exp_pi_mul_I, not_continuousAt_Gamma_neg_nat, I_mul_I, div_I, sin_eq_tsum, SlashInvariantForm.slash_S_apply, exp_sub_pi_mul_I, SchwartzMap.fderivCLM_fourier_eq, hasSum_taylorSeries_neg_log, iteratedDeriv_odd_cos, I_sq, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add, HurwitzZeta.sinZeta_neg, ZMod.LFunction_apply_zero_of_even, digamma_one, VectorFourier.fourierPowSMulRight_apply, PeriodPair.derivWeierstrassPExcept_neg, hasDerivAt_Gamma_one, sin_sub_pi, EisensteinSeries.qExpansion_identity, iteratedDeriv_even_sin, GaussianFourier.integral_cexp_neg_sum_mul_add, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, LSeries.term_def₀, HurwitzZeta.hasSum_int_sinZeta, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, cos_sub_cos, ArithmeticFunction.LSeries_zeta_eulerProduct_exp_log, cos_eq_zero_iff_sin_eq, UpperHalfPlane.im_inv_neg_coe_pos, hasDerivAt_Gamma_one_half, Real.tendsto_integral_gaussian_smul, sin_neg, bernoulliFourierCoeff_recurrence, DirichletCharacter.Even.LFunction_neg_two_mul_nat_add_one, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, cosh_neg, sin_two_pi_sub, VectorFourier.fourierIntegral_probChar, integral_Ioi_cpow_of_lt, HurwitzZeta.sinKernel_def, hasSum_taylorSeries_log, ProbabilityTheory.IsGaussian.charFunDual_eq_of_integral_eq_zero, norm_exp_mul_exp_add_exp_neg_le_of_abs_im_le, integral_gaussian_complex, Real.hasFDerivAt_fourierChar_neg_bilinear_right, summable_bernoulli_fourier, unitary.norm_sub_one_lt_two_iff, GaussianFourier.integrable_cexp_neg_sum_mul_add, HurwitzZeta.jacobiTheta₂''_conj, DirichletCharacter.Odd.LFunction_neg_two_mul_nat_sub_one, Real.iteratedDeriv_fourierIntegral, HurwitzZeta.hurwitzZetaOdd_neg, cos_eq_neg_one_iff, logDeriv_cos, arg_eq_nhds_of_re_neg_of_im_pos, log_neg_I, jacobiTheta₂_add_left', hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, riemannZeta_neg_nat_eq_bernoulli', riemannZeta_eulerProduct_tprod, neg_ofReal_mem_slitPlane, LSeries_hasDerivAt, ofReal_comp_neg, Real.hasDerivAt_fourierIntegral, conj_neg_I, fourier_add_half_inv_index, fourierCoeffOn_of_hasDerivAt, integrable_cexp_quadratic, HurwitzZeta.completedHurwitzZetaEven_residue_zero, ArithmeticFunction.iteratedDeriv_LSeries_alternating, cos_nat_mul_two_pi_add_pi, exp_mul_I_antiperiodic, arg_eq_nhds_of_re_neg_of_im_neg, hasSum_arctan, riemannZeta_neg_nat_eq_bernoulli, SlashInvariantForm.neg_apply, integrable_mul_cexp_neg_mul_sq, hasDerivAt_log_sub_logTaylor, ZMod.dft_odd_iff, PeriodPair.weierstrassP_neg, riemannZeta_two_mul_nat, NumberField.mixedEmbedding.convexBodyLT'_neg_mem, UpperHalfPlane.im_pnat_div_pos, hasSum_cos, logTaylor_succ, HurwitzZeta.hurwitzZetaOdd_one_sub, HurwitzZeta.hasSum_int_completedSinZeta, eqOn_iteratedDerivWithin_cotTerm_integerComplement, riemannZeta_eulerProduct_hasProd, DirichletCharacter.LSeries_eulerProduct, Gamma_eq_zero_iff, integral_cpow_mul_exp_neg_mul_Ioi, normSq_neg, ModularForm.summable_eta_q, HurwitzZeta.sinZeta_one_sub, Real.fderiv_fourierChar_neg_bilinear_left_apply, LSeriesHasSum.neg, arg_neg_eq_arg_sub_pi_iff, integrable_cexp_neg_mul_sq, conj_I, fourierCoeffOn_of_hasDeriv_right, mem_slitPlane_or_neg_mem_slitPlane, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, neg_im, LSeries.hasDerivAt_term, LSeries_eq_mul_integral', fourier_gaussian_pi, IsPrimitiveRoot.arg_eq_pi_iff, sinh_sub_cosh, continuousAt_gaussian_integral, Real.deriv_fourier, DirichletCharacter.continuousOn_neg_logDeriv_LFunctionTrivChar₁, cosh_sub_sinh, HasStrictDerivAt.ccos, EulerProduct.exp_tsum_primes_log_eq_tsum, cos_pi, fourierIntegral_gaussian_pi', integral_gaussian_complex_Ioi, mellin_comp_mul_left, sin_add_pi, two_sin, hasDerivAt_Gamma_nat, mellin_comp_mul_right, arg_neg_coe_angle, two_cos, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq, LSeriesSummable.neg
instOne 📖	CompOp	513 math math: HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip_eps, HurwitzZeta.hasSum_int_hurwitzZetaOdd, CStarAlgebra.preimage_inr_Icc_zero_one, log_one, Polynomial.Chebyshev.S_two_mul_complex_cos, hasSum_mellin_pi_mul_sq', cos_eq_tsum, norm_log_one_sub_inv_add_logTaylor_neg_le, HurwitzZeta.differentiableAt_hurwitzZetaEven_sub_one_div, EulerSine.integral_cos_mul_cos_pow, one_add_tan_sq_mul_cos_sq_eq_one, LSeries.abscissaOfAbsConv_one, log_neg_one, sin_eq_zero_iff_cos_eq, completedZeta_eq_tsum_of_one_lt_re, NumberField.mixedEmbedding.logMap_one, HurwitzZeta.differentiableAt_hurwitzZeta_sub_one_div, MeasureTheory.charFun_eq_fourierIntegral', PeriodPair.hasFPowerSeriesAt_weierstrassPExcept, PeriodPair.hasSumLocallyUniformly_weierstrassP, HurwitzZeta.hasSum_int_cosKernel₀, HurwitzZeta.hasSum_int_hurwitzZetaEven, norm_sub_one_sq_eq_of_norm_eq_one, ofReal_eq_one, DirichletCharacter.LSeries_eulerProduct_tprod, HurwitzZeta.cosZeta_two_mul_nat', Gamma_nat_eq_factorial, digamma_apply_add_one, cos_eq_one_iff, CStarAlgebra.inr_mem_Icc_iff_norm_le, HurwitzZeta.sinZeta_neg_two_mul_nat_sub_one, ZMod.LFunction_one_sub, one_add_cpow_hasFPowerSeriesAt_zero, tsum_exp_neg_mul_int_sq, deriv_cpow_const, BoundedContinuousFunction.char_zero_eq_one, DirichletCharacter.LSeries_modOne_eq, HurwitzZeta.jacobiTheta₂''_add_left, angle_exp_one, hasSum_one_div_nat_pow_mul_fourier, betaIntegral_scaled, cos_eq_zero_iff, WeakFEPair.symm_Λ₀_eq, HasProdUniformlyOn_sineTerm_prod_on_compact, HurwitzZeta.cosZeta_two_mul_nat, sin_eq_neg_one_iff, hasStrictDerivAt_tan, HurwitzZeta.cosZeta_neg_two_mul_nat_add_one, sqrt_I, fourier_gaussian_pi', MeasureTheory.charFun_eq_fourierIntegral, jacobiTheta₂_functional_equation, HasStrictDerivAt.cpow, Real.norm_exp_I_mul_ofReal_sub_one_le, Gamma_add_one, cpow_neg_one, norm_one_add_mul_inv_le, tendsto_tsum_powerSeries_nhdsWithin_stolzCone, HurwitzZeta.jacobiTheta₂'_functional_equation', arg_one, hasSum_nat_jacobiTheta, Gamma_one, DirichletCharacter.norm_LSeries_product_ge_one, ArithmeticFunction.LSeries_zeta_mul_Lseries_moebius, digamma_one_half, HarmonicAt.analyticAt_complex_partial, Real.nnnorm_exp_I_mul_ofReal_sub_one_le, tendsto_mul_log_one_add_of_tendsto, ModularForm.tsum_logDeriv_eta_q, norm_log_one_add_le, EisensteinSeries.summable_left_one_div_linear_sub_one_div_linear, coe_orthonormalBasisOneI, HasDerivAt.cpow, DirichletCharacter.LFunction_changeLevel, SlashInvariantForm.one_coe_eq_one, cos_eq_two_mul_tan_half_div_one_sub_tan_half_sq, DirichletCharacter.IsPrimitive.completedLFunction_one_sub, EisensteinSeries.E2_slash_action, one_div_Gamma_eq_self_mul_one_div_Gamma_add_one, cos_nat_mul_two_pi_sub_pi, hasDerivAt_GammaIntegral, deriv_ofReal_cpow_const, LSeries_iteratedDeriv, PeriodPair.ite_eq_one_sub_sq_mul_weierstrassP, exp_zero, ArithmeticFunction.not_LSeriesSummable_moebius_at_one, starConvex_one_slitPlane, hasStrictFDerivAt_cpow', InnerProductSpace.laplacianWithin_eq_iteratedFDerivWithin_complexPlane, cos_eq_iff_quadratic, riemannZeta_one_sub, PeriodPair.weierstrassP_bound, cos_nat_mul_two_pi, exp_eq_exp_iff_exp_sub_eq_one, NumberField.tendsto_sub_one_mul_dedekindZeta_nhdsGT, PeriodPair.hasFPowerSeriesAt_derivWeierstrassPExcept, betaIntegral_eval_one_right, tendsto_one_add_cpow_exp_of_tendsto, HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re, betaIntegral_recurrence, cosh_sq_sub_sinh_sq, deriv_Gamma_add_one, HurwitzZeta.hurwitzZetaEven_apply_zero, ArithmeticFunction.vonMangoldt.continuousOn_LFunctionResidueClassAux', HurwitzZeta.expZeta_one_sub, HurwitzZeta.cosZeta_apply_zero, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, Unitary.norm_sub_one_lt_two_iff, fourierIntegral_gaussian_pi, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, norm_exp_I_mul_ofReal_sub_one, cot_series_rep', norm_log_sub_logTaylor_le, hasDerivAt_Gammaℝ_one, cos_two_mul, tendsto_sub_mul_tsum_nat_cpow, fourier_zero, partialGamma_add_one, DirichletCharacter.LSeries_changeLevel, HurwitzZeta.hasSum_nat_hurwitzZetaEven, integral_cpow, BoundedContinuousFunction.innerProbChar_zero, HurwitzZeta.hurwitzZeta_neg_nat, norm_log_one_add_sub_self_le, EisensteinSeries.tendsto_zero_inv_linear_sub, hasStrictFDerivAt_cpow, tendsto_riemannZeta_sub_one_div, ZetaAsymptotics.tendsto_Gamma_term_aux, ZMod.LFunction_neg_two_mul_nat_sub_one, LSeries_eq_mul_integral_of_nonneg, HasFDerivWithinAt.cpow, tendsto_one_add_div_pow_exp, ArithmeticFunction.one_eq_delta, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, riemannZeta_eulerProduct_exp_log, ArithmeticFunction.LSeries_zeta_eq, iteratedDeriv_odd_sin, HarmonicAt.differentiableAt_complex_partial, tendsto_logDeriv_euler_sin_div, AddChar.forall_apply_eq_zero, UnitAddTorus.mFourier_zero, logDeriv_sin_div_eq_cot, HurwitzZeta.cosZeta_one_sub, EulerSine.sin_pi_mul_eq, jacobiTheta_S_smul, zeta_eq_tsum_one_div_nat_cpow, WeakFEPair.Λ₀_eq, exp_nat_mul_two_pi_mul_I, tan_sub', deriv_Gamma_nat, logDeriv_exp, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, DirichletCharacter.LSeries.mul_mu_eq_one, nhdsWithin_lt_le_nhdsWithin_stolzSet, HurwitzZeta.completedHurwitzZetaEven_residue_one, HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one, sinh_sq, continuousAt_Gamma_one, arg_one_add_mem_Ioo, PeriodPair.weierstrassPExcept_def, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, tendsto_norm_tan_atTop, hasDerivAt_Gammaℂ_one, hasSum_sin, tendsto_self_mul_Gamma_nhds_zero, riemannZeta_neg_two_mul_nat_add_one, ModularForm.eta_q_eq_cexp, ArithmeticFunction.convolution_vonMangoldt_const_one, HasDerivWithinAt.cpow, tendsto_nat_mul_log_one_add_of_tendsto, EisensteinSeries.q_expansion_riemannZeta, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, norm_jacobiTheta_sub_one_le, ZMod.LFunction_eq_completed_div_gammaFactor_odd, tan_add', one_add_cpow_hasFPowerSeriesAt_zero, DirichletCharacter.LFunctionTrivChar_eq_mul_riemannZeta, HurwitzZeta.completedHurwitzZetaEven_eq, sin_eq_sin_iff, isBigO_riemannZeta_sub_one_div, hasFDerivAt_cpow, ofReal_one, fourierIntegral_eq_half_sub_half_period_translate, HasProdLocallyUniformlyOn_euler_sin_prod, ModularForm.one_coe_eq_one, ModularForm.differentiableAt_eta_tprod, HadamardThreeLines.interpStrip_eq_of_pos, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, one_div_sub_sq_hasFPowerSeriesOnBall_zero, DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow, PeriodPair.hasSumLocallyUniformly_weierstrassPExcept, completedRiemannZeta_eq, cosh_zero, log_sub_self_isBigO, differentiableAt_complex_iff_differentiableAt_real, ModularForm.logDeriv_one_sub_mul_cexp_comp, cosh_sq, BoundedContinuousFunction.probCharDual_zero, cos_int_mul_two_pi_add_pi, eqOn_iteratedDeriv_cotTerm, exp_list_sum, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, IsPrimitiveRoot.arg_eq_zero_iff, fourier_eval_zero, Polynomial.pow_eq_one_of_mahlerMeasure_eq_one, tendsto_euler_sin_prod, ZMod.completedLFunction_eq, zero_cpow_eq_iff, HurwitzZeta.completedSinZeta_one_sub, EulerSine.integral_sin_mul_sin_mul_cos_pow_eq, HurwitzZeta.hurwitzZeta_residue_one, isBigO_at_im_infty_jacobiTheta_sub_one, tsum_exp_neg_quadratic, riemannZeta_zero, cot_pi_eq_exp_ratio, I_pow_four, HurwitzZeta.differentiableAt_one_completedHurwitzZetaEven_sub_completedHurwitzZetaEven, UpperHalfPlane.denom_J, HurwitzZeta.hurwitzZetaEven_one_sub, HurwitzZeta.completedCosZeta₀_one_sub, hasMellin_one_Ioc, cot_series_rep, Summable.clog_one_sub, HurwitzZeta.completedCosZeta_residue_zero, HurwitzZeta.completedHurwitzZetaEven_one_sub, abel_aux, sin_eq_one_iff, hasDerivAt_logTaylor, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, integral_cexp_quadratic, GammaSeq_add_one_left, HurwitzZeta.sinZeta_two_mul_nat_add_one', cos_bound, HasStrictDerivAt.cpow_const, LSeries_one_eq_riemannZeta, iteratedDeriv_even_cos, sin_sq_add_cos_sq, one_div_sub_pow_hasFPowerSeriesOnBall_zero, QuadraticForm.equivalent_sum_squares, ModularForm.logDeriv_one_sub_cexp, HurwitzZeta.hasSum_nat_completedSinZeta, LSeries_eq_mul_integral, Gammaℂ_add_one, Polynomial.X_pow_sub_X_sub_one_irreducible_aux, LSeries.term_delta, ArithmeticFunction.LSeries_vonMangoldt_eq, EisensteinSeries.qExpansion_identity_pnat, sqrt_neg_one, zeta_eq_tsum_one_div_nat_add_one_cpow, cpow_one, Gammaℝ_one, mem_slitPlane_of_norm_lt_one, PeriodPair.hasSum_weierstrassP, ball_one_subset_slitPlane, tendsto_one_add_div_cpow_exp, Polynomial.Chebyshev.U_complex_cosh, LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_and_nonneg, jacobiTheta₂'_functional_equation, riemannZeta_one, cos_sq_add_sin_sq, HurwitzZeta.hurwitzOddFEPair_ε, MeasureTheory.measureReal_abs_gt_le_integral_charFun, sin_pi_div_two, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, GammaSeq_eq_betaIntegral_of_re_pos, rotationOf_coe, hasSum_arctan_aux, deriv_tan, cos_zero, EisensteinSeries.tendsto_zero_inv_linear, arg_neg_one, DirichletCharacter.deriv_LFunctionTrivChar₁_apply_of_ne_one, DirichletCharacter.Odd.gammaFactor_def, tendsto_logDeriv_euler_cot_sub, GammaIntegral_one, riemannZeta_eulerProduct, DirichletCharacter.norm_LFunction_product_ge_one, MeasureTheory.measureReal_abs_dual_gt_le_integral_charFunDual, LSeriesSummable_one_iff, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, betaIntegral_convergent, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, cos_int_mul_two_pi_sub_pi, ModularForm.summable_logDeriv_one_sub_eta_q, MeasureTheory.norm_one_sub_charFun_le_two, ZMod.LFunction_neg_two_mul_nat_add_one, multipliable_sineTerm, LSeries.iteratedDeriv_alternating, LSeries_one_mul_Lseries_moebius, DirichletCharacter.LSeries_eulerProduct_hasProd, Real.enorm_exp_I_mul_ofReal_sub_one_le, EisensteinSeries.q_expansion_bernoulli, cotTerm_identity, GammaAux_recurrence1, LSeries_delta, InnerProductSpace.laplacian_eq_iteratedFDeriv_complexPlane, norm_sub_one_sq_eqOn_sphere, sqrt_neg_I, HurwitzZeta.sinZeta_two_mul_nat_add_one, sqrt_eq_real_add_ite, Polynomial.Chebyshev.S_two_mul_complex_cosh, LogDeriv_exp, completedRiemannZeta₀_one, fourierCoeffOn_of_hasDerivAt_Ioo, Circle.coe_eq_one, eq_zero_cpow_iff, DirichletCharacter.LSeries_eulerProduct_exp_log, ZMod.completedLFunction_one_sub_odd, HadamardThreeLines.interpStrip_eq_of_mem_verticalStrip, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, mellin_convergent_iff_norm, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat_sub_one, LSeriesHasSum_one, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, sin_eq_two_mul_tan_half_div_one_add_tan_half_sq, fourierIntegral_gaussian, norm_exp_sub_one_le, LSeries.tendsto_cpow_mul_atTop, EisensteinSeries.tendsto_double_sum_S_act, integrable_pow_mul_norm_one_add_mul_inv, HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc, differentiableAt_Gamma_nat_add_one, HurwitzZeta.hurwitzZeta_one_sub, one_cpow, exp_pi_mul_I, hasDerivAt_ofReal_cpow_const, I_mul_I, sin_eq_tsum, Gammaℝ_one_sub_mul_Gammaℝ_one_add, MeasureTheory.measureReal_abs_inner_gt_le_integral_charFun, one_add_cpow_hasFPowerSeriesOnBall_zero, hasStrictDerivAt_cpow_const, GammaSeq_eq_approx_Gamma_integral, hasSum_taylorSeries_neg_log, NumberField.mixedEmbedding.logMap_real, euler_sineTerm_tprod, iteratedDeriv_odd_cos, I_sq, Polynomial.Chebyshev.U_complex_cos, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, HasStrictFDerivAt.cpow, deriv_cpow_const, HurwitzZeta.completedCosZeta_eq, HurwitzZeta.completedCosZeta_one_sub, summable_one_div_nat_cpow, exp_rat_mul_pi_mul_I_pow_two_mul_den, logDeriv_prod_sineTerm_eq_sum_cotTerm, differentiableAt_Gamma_one, digamma_one, DirichletCharacter.not_LSeriesSummable_at_one, hasDerivAt_Gamma_one, PeriodPair.hasSum_weierstrassPExcept, HadamardThreeLines.eventuallyle, completedRiemannZeta₀_one_sub, GammaSeq_mul, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, EisensteinSeries.qExpansion_identity, iteratedDeriv_even_sin, GaussianFourier.integral_cexp_neg_sum_mul_add, tendsto_one_add_pow_exp_of_tendsto, log_sub_logTaylor_isBigO, angle_one_left, fourier_zero', mul_inv_cancel, UpperHalfPlane.denom_one, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, summable_log_one_add_of_summable, ArithmeticFunction.LSeries_zeta_eulerProduct_exp_log, cos_eq_zero_iff_sin_eq, jacobiTheta_eq_tsum_nat, cpow_def, nhdsWithin_stolzCone_le_nhdsWithin_stolzSet, angle_one_right, hasDerivAt_ofReal_cpow_const', hasDerivAt_Gamma_one_half, norm_log_one_sub_inv_sub_self_le, GammaIntegral_add_one, bernoulliFourierCoeff_recurrence, DirichletCharacter.Even.LFunction_neg_two_mul_nat_add_one, ModularForm.is_invariant_one', Gamma_mul_Gamma_add_half, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, integral_Ioi_cpow_of_lt, DirichletCharacter.LFunctionTrivChar_isBigO_near_one_horizontal, cot_eq_exp_ratio, LSeries.pow_mul_term_eq, norm_exp_sub_one_sub_id_le, cos_sq, hasSum_taylorSeries_log, DirichletCharacter.LFunctionTrivChar_residue_one, hasDerivAt_tan, integral_gaussian_complex, tan_sq_div_one_add_tan_sq, CStarAlgebra.inr_mem_Icc_iff_nnnorm_le, completedRiemannZeta_residue_one, unitary.norm_sub_one_lt_two_iff, DirichletCharacter.rootNumber_modOne, completedRiemannZeta_one, DirichletCharacter.Odd.LFunction_neg_two_mul_nat_sub_one, Gammaℝ_div_Gammaℝ_one_sub, EisensteinSeries.summable_right_one_div_linear_sub_one_div_linear_succ, cos_eq_neg_one_iff, HasDerivWithinAt.cpow_const, Gamma_mul_Gamma_one_sub, EisensteinSeries.summable_linear_right_add_one_mul_linear_right, jacobiTheta₂_add_left, betaIntegral_convergent_left, hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, riemannZeta_neg_nat_eq_bernoulli', one_add_cpow_hasFPowerSeriesOnBall_zero, betaIntegral_eval_nat_add_one_right, ZMod.LFunction_residue_one, riemannZeta_eulerProduct_tprod, exp_two_pi_mul_I, ZMod.completedLFunction_one_sub_even, norm_log_one_add_half_le_self, hasMellin_cpow_Ioc, sin_sq, integral_mul_cpow_one_add_sq, riemannZeta_residue_one, one_div_one_sub_hasFPowerSeriesOnBall_zero, fourierCoeffOn_of_hasDerivAt, HurwitzZeta.completedHurwitzZetaEven_residue_zero, ArithmeticFunction.iteratedDeriv_LSeries_alternating, HurwitzZeta.tendsto_hurwitzZeta_sub_one_div_nhds_one, inv_one_add_tan_sq, exp_eq_one_iff, cos_nat_mul_two_pi_add_pi, inv_Gammaℝ_one_sub, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, hasSum_arctan, differentiableWithinAt_complex_iff_differentiableWithinAt_real, riemannZeta_neg_nat_eq_bernoulli, DirichletCharacter.LFunction_isBigO_horizontal, HasFDerivAt.cpow, GammaIntegral_convergent, one_re, riemannZeta_two_mul_nat, HurwitzZeta.completedHurwitzZetaOdd_one_sub, tendsto_euler_sin_prod', coe_basisOneI, multipliableUniformlyOn_euler_sin_prod_on_compact, hasSum_cos, logTaylor_succ, HurwitzZeta.hurwitzZetaOdd_one_sub, Quaternion.coeComplex_one, HurwitzZeta.hasSum_int_completedSinZeta, eqOn_iteratedDerivWithin_cotTerm_integerComplement, zeta_nat_eq_tsum_of_gt_one, HurwitzZeta.hasSum_nat_hurwitzZetaOdd_of_mem_Icc, jacobiTheta₂'_add_left, AddChar.expect_apply_eq_ite, riemannZeta_eulerProduct_hasProd, DirichletCharacter.LSeries_eulerProduct, logDeriv_sineTerm_eq_cotTerm, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, cpow_zero, WeakFEPair.f_modif_aux2, cos_sq', integral_cpow_mul_exp_neg_mul_Ioi, norm_sub_one_sq_eq_of_norm_one, Circle.coe_one, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, HurwitzZeta.sinZeta_one_sub, HurwitzZeta.completedHurwitzZetaEven₀_one_sub, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, approx_Gamma_integral_tendsto_Gamma_integral, tendsto_tsum_powerSeries_nhdsWithin_stolzSet, inv_Gammaℝ_two_sub, Gamma_one_half_eq, cos_int_mul_two_pi, one_mem_slitPlane, cos_two_pi, fourierCoeffOn_of_hasDeriv_right, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_nhds_right, ModularForm.is_invariant_one, exp_int_mul_two_pi_mul_I, GaussianInt.toComplex_one, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, LSeries_eq_mul_integral', Summable.hasSumUniformlyOn_log_one_add, fourier_gaussian_pi, IsPrimitiveRoot.arg_eq_pi_iff, completedRiemannZeta_one_sub, DirichletCharacter.continuousOn_neg_logDeriv_LFunctionTrivChar₁, EulerSine.integral_cos_mul_cos_pow_even, LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div, Gammaℂ_one, one_div_sub_hasFPowerSeriesOnBall_zero, EulerProduct.exp_tsum_primes_log_eq_tsum, cos_pi, Gammaℝ_mul_Gammaℝ_add_one, fourierIntegral_gaussian_pi', ModularForm.multipliableLocallyUniformlyOn_eta, tan_two_mul, integral_gaussian_complex_Ioi, one_im, PeriodPair.weierstrassPExcept_add, multipliable_one_add_of_summable, hasDerivAt_Gamma_nat, tan_eq_one_sub_tan_half_sq_div_one_add_tan_half_sq, normSq_one, sqrt_one, Summable.tendstoUniformlyOn_tsum_nat_log_one_add, HurwitzZeta.tendsto_hurwitzZetaEven_sub_one_div_nhds_one, HasDerivAt.cpow_const, HurwitzZeta.hurwitzZetaEven_residue_one, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq, fourierCoeff_fourier
instRatCast 📖	CompOp	31 math math: nnnorm_ratCast, normSq_ratCast, isPrimitiveRoot_exp_rat, ratCast_im, NumberField.canonicalEmbedding.integralBasis_repr_apply, div_ratCast_re, isIntegral_exp_rat_mul_pi_mul_I, ratCast_re, deriv_Gamma_nat, isAlgebraic_sin_rat_mul_pi, isIntegral_two_mul_sin_rat_mul_pi, norm_ratCast, isPrimitiveRoot_exp_rat_of_even_num, isIntegral_exp_neg_rat_mul_pi_mul_I, NumberField.inverse_basisMatrix_mulVec_eq_repr, EisensteinSeries.q_expansion_bernoulli, div_ratCast_im, isPrimitiveRoot_exp_rat_of_odd_num, isIntegral_two_mul_cos_rat_mul_pi, exp_rat_mul_pi_mul_I_pow_two_mul_den, ofReal_ratCast, isAlgebraic_tan_rat_mul_pi, riemannZeta_neg_nat_eq_bernoulli', PeriodPair.mul_ω₁_add_mul_ω₂_mem_lattice, NumberField.canonicalEmbedding_eq_basisMatrix_mulVec, isAlgebraic_cos_rat_mul_pi, riemannZeta_neg_nat_eq_bernoulli, riemannZeta_two_mul_nat, NumberField.house.basis_repr_norm_le_const_mul_house, div_ratCast, hasDerivAt_Gamma_nat
instRepr 📖	CompOp	—
instRing 📖	CompOp	131 math math: NumberField.basisMatrix_eq_embeddingsMatrixReindex, TemperedDistribution.lineDerivOpCLM_eq, NumberField.mixedEmbedding.fundamentalDomain_integerLattice, HasFDerivWithinAt.ccosh, HasFDerivWithinAt.ccos, PeriodPair.hasFPowerSeriesAt_weierstrassPExcept, Real.hasFDerivAt_fourierChar_neg_bilinear_left, Ideal.toCharacterSpace_apply_eq_zero_of_mem, lim_conj, NumberField.canonicalEmbedding.mem_span_latticeBasis, HasStrictFDerivAt.csin, instIsComplete, fderivWithin_csinh, span_fourier_closure_eq_top, one_add_cpow_hasFPowerSeriesAt_zero, UnitAddTorus.mFourierSubalgebra_coe, exp_eq_exp_ℂ, hasStrictFDerivAt_exp_real, sum_cauchyPowerSeries_eq_integral, HasFDerivWithinAt.csin, gelfandStarTransform_naturality, isCauSeq_conj, gelfandStarTransform_apply_apply, exp_def, fderivWithin_ccosh, Polynomial.cyclotomic.eval_apply_ofReal, TemperedDistribution.smulLeftCLM_neg, CStarMatrix.norm_def', LindemannWeierstrass.exp_polynomial_approx, hasStrictFDerivAt_cpow', PeriodPair.hasFPowerSeriesAt_derivWeierstrassPExcept, CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM, spectrum.gelfandTransform_eq, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, NumberField.canonicalEmbedding.integerLattice.inter_ball_finite, Polynomial.logMahlerMeasure_X_sub_C, NumberField.mixedEmbedding.fundamentalDomain_idealLattice, gelfandStarTransform_symm_apply, hasStrictFDerivAt_cpow, HasFDerivWithinAt.cpow, HasStrictFDerivAt.const_cpow, HasFDerivWithinAt.csinh, isIntegral_exp_rat_mul_pi_mul_I, HasStrictFDerivAt.cexp, StarAlgebra.elemental.bijective_characterSpaceToSpectrum, fderivWithin_ccos, HasStrictFDerivAt.clog, isAlgebraic_sin_rat_mul_pi, HasDerivWithinAt.complexToReal_fderiv, HasStrictFDerivAt.ccos, HasFDerivWithinAt.cexp, HasDerivAt.complexToReal_fderiv, liftAux_I, TemperedDistribution.smulLeftCLM_sub, one_add_cpow_hasFPowerSeriesAt_zero, hasFDerivAt_cpow, NormedSpace.ofReal_exp_ℝ_ℝ, HasFDerivAt.cexp, HasFDerivAt.ccos, HasFDerivAt.clog, one_div_sub_sq_hasFPowerSeriesOnBall_zero, restrictScalars_one_smulRight, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, Algebra.norm_complex_eq, HasFDerivAt.ccosh, isIntegral_two_mul_sin_rat_mul_pi, PositiveLinearMap.gnsStarAlgHom_apply, instNonemptyElemWeakDualComplexCharacterSpaceOfNontrivial, fderiv_ccos, one_div_sub_pow_hasFPowerSeriesOnBall_zero, isIntegral_exp_neg_rat_mul_pi_mul_I, restrictScalars_toSpanSingleton, HasFDerivWithinAt.clog, fourierSubalgebra_coe, Polynomial.toAddCircle_monomial_eq_smul_fourier, Polynomial.mahlerMeasure_X_sub_C, Algebra.norm_complex_apply, isConformalMap_complex_linear_conj, UnitAddTorus.hasSum_mFourier_series_of_summable, HasFDerivAt.csin, WStarAlgebra.exists_predual, HasFDerivWithinAt.const_cpow, hasFPowerSeriesOnBall_cuspFunction, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, fderivWithin_csin, WeakDual.CharacterSpace.mem_spectrum_iff_exists, lim_im, isIntegral_two_mul_cos_rat_mul_pi, one_add_cpow_hasFPowerSeriesOnBall_zero, hasSum_fourier_series_of_summable, NormedSpace.of_real_exp_ℝ_ℝ, fderiv_ccosh, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, HasStrictFDerivAt.cpow, isCauSeq_im, lim_norm, UnitAddTorus.span_mFourier_closure_eq_top, Real.hasFDerivAt_fourierChar_neg_bilinear_right, equiv_limAux, isAlgebraic_tan_rat_mul_pi, hasStrictFDerivAt_log_real, isIntegral_rat_I, fderiv_csinh, one_add_cpow_hasFPowerSeriesOnBall_zero, HasFDerivAt.csinh, HasStrictFDerivAt.ccosh, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, VonNeumannAlgebra.instSubringClass, one_div_one_sub_hasFPowerSeriesOnBall_zero, isConformalMap_complex_linear, fderiv_csin, Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero, isAlgebraic_cos_rat_mul_pi, HasFDerivAt.cpow, HasStrictDerivAt.complexToReal_fderiv, WeakDual.CharacterSpace.exists_apply_eq_zero, Polynomial.toAddCircle_C_eq_smul_fourier_zero, isCauSeq_exp, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, HasStrictFDerivAt.csinh, cfcHom_eq_of_isStarNormal, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, isIntegral_int_I, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, continuousFunctionalCalculus_map_id, isCauSeq_re, WeakDual.CharacterSpace.instStarHomClass, one_div_sub_hasFPowerSeriesOnBall_zero, lim_re, HasFDerivAt.const_cpow, lim_eq_lim_im_add_lim_re
instSemiring 📖	CompOp	1074 math math: Polynomial.logMahlerMeasure_one, CuspFormClass.qExpansion_isBigO, Real.hasDerivAt_fourier, NumberField.basisMatrix_eq_embeddingsMatrixReindex, TemperedDistribution.lineDerivOpCLM_eq, MeasureTheory.Lp.toTemperedDistributionCLM_apply, TemperedDistribution.fourierInv_toTemperedDistributionCLM_eq, cos_eq_tsum, NumberField.InfinitePlace.Completion.isometry_extensionEmbeddingOfIsReal, NumberField.InfinitePlace.isNontrivial, NumberField.InfinitePlace.eq_iff_isEquiv, Polynomial.mahlerMeasure_const, Polynomial.mahlerMeasure_def_of_ne_zero, EulerSine.integral_cos_mul_cos_pow, one_add_tan_sq_mul_cos_sq_eq_one, ModularForm.smul_slash, GaussianInt.toComplex_mul, restrictScalars_one_smulRight', coe_realPart, UnitDisc.normSq_lt_one, SchwartzMap.integral_sesq_fourier_fourier, GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I, cos_three_mul, EisensteinSeries.hasSum_e2Summand_symmetricIcc, normSq_pos, ModularFormClass.qExpansionFormalMultilinearSeries_apply_norm, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, NumberField.canonicalEmbedding.mem_span_latticeBasis, PeriodPair.hasSumLocallyUniformly_weierstrassP, NumberField.mixedEmbedding.norm_eq_sup'_normAtPlace, circleTransformDeriv_eq, NumberField.mixedEmbedding.convexBodyLT'_mem, NumberField.InfinitePlace.comap_mk, ofRealCLM_coe, CliffordAlgebraComplex.ofComplex_conj, GaussianInt.intCast_im, CStarModule.innerₛₗ_apply, summable_riemannZetaSummand, normSq_natCast, Real.fourierIntegralInv_eq', PositiveLinearMap.apply_le_of_isSelfAdjoint, ZMod.LFunction_stdAddChar_eq_expZeta, GaussianInt.toComplex_def', integral_boundary_rect_eq_zero_of_differentiableOn, qExpansionRingHom_apply, LSeries.term_convolution', PeriodPair.derivWeierstrassPExcept_sub, Polynomial.mahlerMeasure_eq_leadingCoeff_mul_prod_roots, isLittleO_exp_neg_mul_sq_cocompact, norm_jacobiTheta₂_term_fderiv_ge, UpperHalfPlane.normSq_pos, NumberField.canonicalEmbedding.latticeBasis_apply, HurwitzZeta.cosZeta_two_mul_nat', norm_mul_self_eq_normSq, NumberField.InfinitePlace.le_iff_le, real_algHom_eq_id_or_conj, summableLocallyUniformlyOn_iteratedDerivWithin_cexp, NumberField.norm_norm_le_norm_mul_house_pow, realPart_ofReal, span_fourier_closure_eq_top, Real.fourierIntegral_eq, ModularForm.eta_q_eq_pow, HasDerivWithinAt.complexToReal_fderiv', circleIntegral_def_Icc, ZMod.LFunction_one_sub, hasSum_taylorSeries_of_entire, NumberField.isInfinitePlace_iff, circleIntegrable_iff, tsum_exp_neg_mul_int_sq, NumberField.InfinitePlace.isometry_embedding, ZMod.dft_even_iff, Unitization.real_cfcₙ_eq_cfc_inr, NumberField.InfinitePlace.Completion.surjective_extensionEmbedding_of_isComplex, ZMod.invDFT_apply, NumberField.mixedEmbedding.normAtAllPlaces_apply, imaginaryPart_I_smul, NumberField.mem_maximalRealSubfield_iff, Polynomial.logMahlerMeasure_C_mul, SchwartzMap.integral_bilin_fourierInv_eq, NumberField.mixedEmbedding.fundamentalCone.integerSetEquivNorm_apply_fst, GaussianFourier.integral_cexp_neg_mul_sq_norm, ArithmeticFunction.LSeriesHasSum_mul, NumberField.mixedEmbedding.normAtPlace_polarCoord_symm_of_isComplex, affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div, hasSum_one_div_nat_pow_mul_fourier, NumberField.mixedEmbedding.norm_unit_smul, two_mul_riemannZeta_eq_tsum_int_inv_pow_of_even, NumberField.mixedEmbedding.norm_eq_norm, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, TemperedDistribution.lineDerivOp_fourier_eq, WeakFEPair.symm_Λ₀_eq, VonNeumannAlgebra.ext_iff, reLm_coe, ProbabilityTheory.hasDerivAt_integral_pow_mul_exp, HurwitzZeta.cosZeta_two_mul_nat, NumberField.mixedEmbedding.mem_idealLattice, re_balance, UpperHalfPlane.im_smul_eq_div_normSq, IsConformalMap.is_complex_or_conj_linear, NumberField.mixedEmbedding.fundamentalCone.completeBasis_apply_of_ne, riemannZeta_two, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, hasStrictDerivAt_tan, fourierIntegral_gaussian_innerProductSpace', fourier_gaussian_pi', NumberField.InfiniteAdeleRing.algebraMap_apply, UnitAddTorus.mFourierSubalgebra_coe, TemperedDistribution.smulLeftCLM_compL_smulLeftCLM, iteratedDerivWithin_tsum_cexp_eq, UpperHalfPlane.σ_sq, jacobiTheta₂_functional_equation, ZMod.dft_mul_const, hasSum_taylorSeries_on_ball, cosh_three_mul, im_expect, GaussianFourier.integral_cexp_neg_mul_sq_norm_add, norm_cexp_neg_mul_sq, UpperHalfPlane.moebius_im, isQuotientCoveringMap_npow, sum_cauchyPowerSeries_eq_integral, CStarModule.innerSL_apply, Real.vector_fourierIntegral_eq_integral_exp_smul, taylorSeries_eq_of_entire, HurwitzZeta.evenKernel_def, UnitAddTorus.mFourierSubalgebra_closure_eq_top, ModularGroup.one_lt_normSq_T_zpow_smul, Real.fourier_real_eq_integral_exp_smul, instFiniteDimensionalReal, CStarAlgebra.span_nonneg_inter_unitBall, normSq_ratCast, tendsto_tsum_powerSeries_nhdsWithin_stolzCone, MeasureTheory.Lp.instFourierInvSMul, norm_pow, tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support, NumberField.mixedEmbedding.commMap_apply_of_isComplex, hasSum_nat_jacobiTheta, NumberField.InfinitePlace.Completion.norm_coe, Real.fourier_deriv, DirichletCharacter.norm_LSeries_product_ge_one, PeriodPair.hasSum_derivWeierstrassPExcept, CliffordAlgebraComplex.ofComplex_toComplex, toMatrix_rotation, normSq_mk, GaussianInt.toComplex_def₂, ZMod.stdAddChar_coe, DifferentiableOn.circleIntegral_sub_inv_smul, ValueDistribution.proximity_pow_zero, CliffordAlgebraComplex.toComplex_comp_ofComplex, gelfandStarTransform_naturality, NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_dvd_norm_le, instPathConnectedSpaceUnits, GaussianInt.normSq_le_normSq_of_re_le_of_im_le, WeakFEPair.h_feq, SchwartzMap.integral_fourierInv_smul_eq, ModularForm.levelOne_neg_weight_rank_zero, CStarMatrix.toCLM_injective, integral_circleTransform, Polynomial.toAddCircle_X_pow_eq_fourier, HurwitzZeta.oddKernel_def', cos_eq_two_mul_tan_half_div_one_sub_tan_half_sq, SchwartzMap.integral_bilin_fourierIntegral_eq, fourier_gaussian_innerProductSpace, GaussianInt.toComplex_div_re, VectorFourier.fourierPowSMulRight_eq_comp, mem_rootsOfUnity, NumberField.mixedEmbedding.convexBodyLT_mem, NumberField.InfinitePlace.coe_apply, TemperedDistribution.smulLeftCLM_neg, NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_norm_eq_mul_torsion, Function.HasTemperateGrowth.toTemperedDistribution_apply, sin_bound, hasDerivAt_GammaIntegral, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, LSeries_iteratedDeriv, NumberField.ComplexEmbedding.conjugate_comp, SchwartzMap.integral_sesq_fourier_eq, CStarMatrix.norm_def', PeriodPair.ite_eq_one_sub_sq_mul_weierstrassP, NumberField.mixedEmbedding.normAtPlace_apply_of_isReal, taylorSeries_eq_on_ball', StrongFEPair.functional_equation, fourierCoeff_bernoulli_eq, UpperHalfPlane.denom_cocycle_σ, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace, circleIntegral.integral_sub_inv_smul_sub_smul, EulerSine.integral_cos_mul_cos_pow_aux, Polynomial.logMahlerMeasure_mul_eq_add_logMahelerMeasure, TemperedDistribution.instLineDerivLeftSMulReal, LindemannWeierstrass.exp_polynomial_approx, CStarMatrix.inner_toCLM_conjTranspose_left, normSq_mul, NumberField.ComplexEmbedding.Extension.not_isReal_of_not_isReal, TemperedDistribution.fourierInv_lineDerivOp_eq, mul_conj', hasStrictFDerivAt_cpow', AddChar.sum_apply_eq_ite, PositiveLinearMap.preGNS_norm_def, PositiveLinearMap.leftMulMapPreGNS_apply, cpow_ofNat_mul', cos_eq_iff_quadratic, liftAux_apply, NumberField.exists_conjugate_one_le_norm, NumberField.InfinitePlace.conjugate_embedding_injective, NumberField.InfinitePlace.norm_embedding_eq, PeriodPair.hasSum_derivWeierstrassP, sinh_three_mul, ext_inner_map, GaussianInt.toComplex_injective, PeriodPair.weierstrassP_bound, TemperedDistribution.instFourierAdd, PeriodPair.hasSumLocallyUniformly_derivWeierstrassP, cauchyPowerSeries_apply, Polynomial.logMahlerMeasure_mul_eq_add_logMahlerMeasure, ContinuousLinearMap.instStarOrderedRing, CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM, TemperedDistribution.instFourierSMul, exp_nsmul, NumberField.InfinitePlace.mkReal_coe, qExpansion_of_mul, cosh_sq_sub_sinh_sq, DiffContOnCl.circleIntegral_sub_inv_smul, CompletelyPositiveMap.instLinearMapClassComplex, LSeriesHasSum.convolution, NumberField.InfinitePlace.Completion.isometry_extensionEmbedding, NumberField.InfinitePlace.smul_mk, BoundedContinuousFunction.mem_charPoly, spectrum.gelfandTransform_eq, BoundedContinuousFunction.charAlgHom_apply, lift_apply, NumberField.ComplexEmbedding.involutive_conjugate, TemperedDistribution.delta_apply, NumberField.InfiniteAdeleRing.mixedEmbedding_eq_algebraMap_comp, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, fourierIntegral_gaussian_pi, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, NumberField.Units.map_complexEmbedding_torsion, Real.circleAverage_eq_circleIntegral, CStarAlgebra.span_nonneg, NumberField.mixedEmbedding.fundamentalCone.norm_pos_of_mem, TemperedDistribution.instContinuousFourier, cos_eq_tsum', mellinInv_eq_fourierIntegralInv, CStarModule.inner_smul_right_complex, NumberField.canonicalEmbedding.integralBasis_repr_apply, PositiveLinearMap.instContinuousLinearMapClassComplexOfLinearMapClassOfOrderHomClass, sq_norm, NumberField.canonicalEmbedding.integerLattice.inter_ball_finite, UpperHalfPlane.σ_mul, NumberField.ComplexEmbedding.IsConj.eq, tendsto_integral_exp_smul_cocompact, UnitDisc.coe_pow, hasDerivAt_Gammaℝ_one, hasFDerivAt_jacobiTheta₂_term, Polynomial.logMahlerMeasure_X_sub_C, gelfandStarTransform_symm_apply, NumberField.InfinitePlace.card_filter_mk_eq, riemannZeta_four, cos_two_mul, Polynomial.mahlerMeasure_eq_zero_iff, ZMod.dft_comp_unitMul, Polynomial.toAddCircle.integrable, normSq_I, CStarModule.inner_smul_left_complex, SchwartzMap.delta_apply, norm_cauchyPowerSeries_le, re_comp_balance, cos_add_sin_mul_I_pow, mellinInv_eq_fourierInv, summable_jacobiTheta₂_term_fderiv_iff, fourier_gaussian_innerProductSpace', VectorFourier.fourierIntegral_const_smul, Real.fourier_real_eq, UpperHalfPlane.norm_σ, hasSum_fourier_series_L2, NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis, tendsto_normSq_cocompact_atTop, NumberField.mixedEmbedding.convexBodySumFun_apply, Real.tendsto_integral_gaussian_smul', HurwitzZeta.hurwitzZeta_neg_nat, GaussianFourier.integrable_cexp_neg_mul_sum_add, hasStrictFDerivAt_cpow, NumberField.InfinitePlace.Completion.locallyCompactSpace, exp_nsmul', Real.fourierIntegral_convergent_iff', AddChar.expect_apply_eq_zero_iff_ne_zero, normSq_add, hasStrictDerivAt_sinh, ofReal_balance, inner_map_polarization', PositiveLinearMap.preGNS_inner_def, NumberField.InfinitePlace.IsRamified.comap_embedding_conjugate, EisensteinSeries.G2_S_transform, re_sq_le_normSq, div_im, tendsto_one_add_div_pow_exp, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, NumberField.InfinitePlace.Completion.isClosed_image_extensionEmbedding, NumberField.Units.complexEmbedding_injective, NumberField.mixedEmbedding.fundamentalCone.mem_normLeOne, realPart_add_I_smul_imaginaryPart, iteratedDeriv_odd_sin, Polynomial.logMahlerMeasure_X_add_C, AddChar.forall_apply_eq_zero, Real.fourierIntegral_iteratedDeriv, CStarAlgebra.span_nonneg_inter_unitClosedBall, EulerSine.sin_pi_mul_eq, qExpansion_one, integral_mul_cexp_neg_mul_sq, NumberField.mixedEmbedding.fundamentalCone.norm_expMapBasis, normSq_eq_norm_sq, VonNeumannAlgebra.centralizer_centralizer', ModularForm.levelOne_weight_zero_rank_one, I_pow_eq_pow_mod, AddChar.sum_apply_eq_zero_iff_ne_zero, norm_exp_mul_sq_le, WeakFEPair.Λ₀_eq, CliffordAlgebraComplex.ofComplex_I, norm_exp_sub_sum_le_norm_mul_exp, NumberField.InfinitePlace.Completion.extensionEmbeddingOfIsReal_coe, Polynomial.intervalIntegrable_mahlerMeasure, PeriodPair.weierstrassPExcept_eq_tsum, liftAux_neg_I, NumberField.mixedEmbedding.indexEquiv_apply_isComplex_fst, spectrum.map_pow, NumberField.InfinitePlace.isInfinitePlace, ProbabilityTheory.complexMGF_gaussianReal, TemperedDistribution.fourier_toTemperedDistributionCLM_eq, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, hasDerivAt_jacobiTheta₂_fst, sinh_sq, NumberField.mixedEmbedding.unit_smul_eq_iff_mul_eq, cpow_nat_mul, VectorFourier.fourierIntegral_convergent_iff, NumberField.InfinitePlace.ComplexEmbedding.conjugate_sign, normSq_sub, GaussianInt.toComplex_im_div, hasDerivAt_sinh, PeriodPair.weierstrassPExcept_def, NumberField.InfinitePlace.smul_coe_apply, inv_re, WeakFEPair.h_feq', isCauSeq_norm_exp, hasSum_jacobiTheta₂_term_fderiv, Polynomial.mahlerMeasure_C_mul_X_add_C, TemperedDistribution.fourierTransformInv_toTemperedDistributionCLM_eq, hasDerivAt_Gammaℂ_one, hasSum_sin, SchwartzMap.integral_bilin_fourier_eq, det_conjAe, hasStrictDerivAt_const_cpow, PositiveLinearMap.toPreGNS_ofPreGNS, NumberField.mixedEmbedding.fundamentalCone.existsUnique_preimage_of_mem_integerSet, EisensteinSeries.hasSum_e2Summand_symmetricIco, ArithmeticFunction.convolution_vonMangoldt_const_one, HasDerivAt.complexToReal_fderiv', circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const, SchwartzMap.toTemperedDistributionCLM_apply_apply, EisensteinSeries.q_expansion_riemannZeta, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, PositiveLinearMap.exists_norm_apply_le, Real.fourierIntegral_real_eq_integral_exp_smul, Real.deriv_fourierIntegral, liftAux_I, fourierSubalgebra_separatesPoints, ArithmeticFunction.convolution_vonMangoldt_zeta, TemperedDistribution.smulLeftCLM_sub, fourierCoeff.const_smul, NumberField.mixedEmbedding.normAtPlace_add_le, hasSum_cauchyPowerSeries_integral, pow_cpow_ofNat_inv, Circle.coe_pow, MeasureTheory.Lp.instFourierSMul, LSeries_convolution', LSeries.term_convolution, hasFDerivAt_cpow, UpperHalfPlane.σ_num, fourierIntegral_eq_half_sub_half_period_translate, algHom_ext_iff, sq_nonneg_iff, VonNeumannAlgebra.mem_carrier, VonNeumannAlgebra.centralizer_centralizer, selfAdjointEquiv_apply, NumberField.IsCMField.complexEmbedding_complexConj, range_liftAux, UpperHalfPlane.normSq_denom_pos, sin_three_mul, Polynomial.one_le_mahlerMeasure_of_ne_zero, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, liftAux_apply_I, PeriodPair.summable_weierstrassPSummand, one_div_sub_sq_hasFPowerSeriesOnBall_zero, PeriodPair.hasSumLocallyUniformly_weierstrassPExcept, UpperHalfPlane.coe_smul, inv_def, equivRealProdLm_apply, CStarAlgebra.linear_combination_nonneg, hasDerivAt_cos, ZMod.invDFT_def', log_sub_self_isBigO, differentiableAt_complex_iff_differentiableAt_real, cfcHom_real_eq_restrict, NumberField.ComplexEmbedding.IsConj.isReal_comp, two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, cosh_sq, PeriodPair.derivWeierstrassP_sq, ArithmeticFunction.LSeries_mul', GaussianInt.toComplex_add, CStarMatrix.toCLM_apply_single, hasSum_two_pi_I_cauchyPowerSeries_integral, normSq_ofNat, Polynomial.logMahlerMeasure_eq_log_leadingCoeff_add_sum_log_roots, Circle.instContinuousSMul, eqOn_iteratedDeriv_cotTerm, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, tsum_riemannZetaSummand, restrictScalars_one_smulRight, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, Algebra.norm_complex_eq, VectorFourier.integral_fourierIntegral_swap, GaussianInt.toComplex_sub, cos_two_mul', tendsto_euler_sin_prod, normSq_ofReal_add_I_mul_sqrt_one_sub, hasStrictDerivAt_log, NumberField.mixedEmbedding.exists_ne_zero_mem_ringOfIntegers_lt', Fourier.fourierIntegral_const_smul, CStarMatrix.toCLM_apply_eq_sum, GaussianInt.toComplex_star, EulerSine.integral_sin_mul_sin_mul_cos_pow_eq, Polynomial.mahlerMeasure_X_add_C, NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm, NumberField.mixedEmbedding.logMap_eq_logEmbedding, conjAe_coe, GaussianInt.toComplex_def, ProbabilityTheory.hasDerivAt_complexMGF, AddChar.card_eq, tsum_exp_neg_quadratic, ModularFormClass.qExpansion_coeff, Polynomial.mahlerMeasure_mul, ZMod.dft_apply, conj_mul', inr_comp_cfcₙHom_eq_cfcₙAux, skewAdjoint.negISMul_apply_coe, DiffContOnCl.circleAverage_smul_div, GaussianInt.to_real_re, cosh_eq_tsum, I_pow_four, rank_real_complex', Polynomial.logMahlerMeasure_def, Polynomial.logMahlerMeasure_X, Polynomial.prod_mahlerMeasure_eq_mahlerMeasure_prod, CStarMatrix.ofMatrix_eq_ofMatrixL, Real.fourier_eq, normSq_zero, isOpenQuotientMap_pow, ZMod.invDFT_apply', NumberField.canonicalEmbedding.nnnorm_eq, imaginaryPart_eq_neg_I_smul_skewAdjointPart, TemperedDistribution.instLineDerivSMulReal, MeasureTheory.Measure.toTemperedDistribution_apply, tendsto_integral_exp_smul_cocompact_of_inner_product, CStarMatrix.toCLM_apply, DirichletCharacter.convolution_twist_vonMangoldt, ofReal_comp_balance, ValueDistribution.characteristic_pow_zero, bernoulliFourierCoeff_eq, summable_dirichletSummand, NumberField.mixedEmbedding.norm_nonneg, cosh_two_mul, NumberField.mixedEmbedding.mixedEmbedding_apply_isReal, normSq_conj, abel_aux, continuous_normSq, TemperedDistribution.fourier_lineDerivOp_eq, equivRealProdLm_symm_apply, span_selfAdjoint, equivRealProdLm_symm_apply_im, coe_basisOneI_repr, realPart_I_smul, hasDerivAt_logTaylor, ModularGroup.tendsto_normSq_coprime_pair, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, NumberField.mixedEmbedding.norm_smul, PositiveLinearMap.gnsStarAlgHom_apply, Real.fourier_iteratedDeriv, normSq_intCast, GaussianInt.to_real_im, NumberField.InfinitePlace.IsRamified.comap_embedding, NumberField.InfinitePlace.LiesOver.embedding_comp_eq_or_conjugate_embedding_comp_eq, WeakFEPair.hf_zero, cpow_mul_nat, LinearMap.isPositive_iff_complex, GaussianInt.toComplex_re_div, VectorFourier.fourierSMulRight_apply, UnitAddTorus.instContinuousSMulComplexSubtypeAEEqFunVolumeMemAddSubgroupLp, integral_cexp_quadratic, TemperedDistribution.lineDerivOp_toTemperedDistributionCLM_eq, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, HurwitzZeta.sinZeta_two_mul_nat_add_one', ModularFormClass.qExpansion_coeff_zero, GaussianInt.toComplex_re, imaginaryPart_apply_coe, cos_bound, UnitAddTorus.hasSum_mFourier_series_L2, cpow_natCast, TemperedDistribution.laplacian_eq_sum, NumberField.mixedEmbedding.stdBasis_repr_eq_matrixToStdBasis_mul, spectrum_subset_unitary_of_mem_unitary, WithCStarModule.pi_inner, NumberField.mixedEmbedding.continuous_norm, NumberField.discr_eq_basisMatrix_det_sq, ZMod.dft_comp_neg, NumberField.mixedEmbedding.det_matrixToStdBasis, im_balance, ofReal_expect, AddChar.starComp_apply, TemperedDistribution.instContinuousLineDeriv, iteratedDeriv_even_cos, div_re, sin_sq_add_cos_sq, NumberField.mixedEmbedding.fundamentalCone.norm_normAtAllPlaces, Real.fourier_bilin_convolution_eq, one_div_sub_pow_hasFPowerSeriesOnBall_zero, CStarMatrix.mul_entry_mul_eq_inner_toCLM, GaussianInt.toComplex_inj, QuadraticForm.equivalent_sum_squares, CliffordAlgebraComplex.equiv_symm_apply, SchwartzMap.integral_sesq_fourierIntegral_eq, Real.fourierIntegral_eq', hasSum_taylorSeries_on_emetric_ball, hasFPowerSeriesOn_cauchy_integral, NumberField.InfinitePlace.embedding_of_isReal_apply, contDiffOn_tsum_cexp, Algebra.leftMulMatrix_complex, NormedRing.algEquivComplexOfComplete_symm_apply, circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable, Polynomial.X_pow_sub_X_sub_one_irreducible_aux, NumberField.mixedEmbedding.normAtAllPlaces_mixedEmbedding, hasStrictDerivAt_cos, Orientation.kahler_apply_self, HasStrictDerivAt.complexToReal_fderiv', fourierIntegral_gaussian_innerProductSpace, cexp_neg_quadratic_isLittleO_abs_rpow_cocompact, WithCStarModule.inner_single_right, EisensteinSeries.qExpansion_identity_pnat, DirichletCharacter.IsPrimitive.fourierTransform_eq_inv_mul_gaussSum, LSeries.abscissaOfAbsConv_convolution_le, Real.iteratedDeriv_fourier, CliffordAlgebraComplex.toComplex_ι, UpperHalfPlane.σ_ofReal, linearEquiv_det_conjAe, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', NumberField.mixedEmbedding.normAtPlace_polarCoord_symm_of_isReal, NumberField.Units.complexEmbedding_apply, PeriodPair.hasSum_weierstrassP, NumberField.count_multisetInfinitePlace_eq_mult, TemperedDistribution.smulLeftCLM_add, restrictScalars_toSpanSingleton, Orientation.normSq_kahler, hasSum_cos', normSq_inv, ZMod.dft_dft, NumberField.InfinitePlace.Completion.extensionEmbedding_coe, AddChar.coe_doubleDualEquiv, fourierSubalgebra_coe, NumberField.InfinitePlace.denseRange_algebraMap_pi, Polynomial.toAddCircle_monomial_eq_smul_fourier, jacobiTheta₂'_functional_equation, NumberField.mixedEmbedding.fundamentalCone.mixedEmbedding_preimageOfMemIntegerSet, NumberField.mixedEmbedding.fundamentalCone.normAtPlace_pos_of_mem, hasStrictDerivAt_sin, Polynomial.sum_sq_norm_coeff_eq_circleAverage, expMonoidHom_apply, Polynomial.mahlerMeasure_X_sub_C, cos_sq_add_sin_sq, exp_sub_sum_range_succ_isLittleO_pow, normSq_div, continuousOn_prod_circle_transform_function, pow_cpow_nat_inv, natCast_cpow_natCast_mul, NumberField.conj_basisMatrix, normSq_ofReal, hasFDerivAt_jacobiTheta₂, im_comp_balance, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, Algebra.norm_complex_apply, deriv_tan, norm_def, NumberField.InfinitePlace.isometry_embedding_of_isReal, NumberField.ComplexEmbedding.IsReal.coe_embedding_apply, hasDerivAt_cosh, Unitization.nnreal_cfcₙ_eq_cfc_inr, cpow_ofNat_mul, NumberField.inverse_basisMatrix_mulVec_eq_repr, TemperedDistribution.fourierTransform_apply, NumberField.InfinitePlace.card_real_embeddings, LSeriesSummable.convolution, isConformalMap_complex_linear_conj, exp_sub_sum_range_isBigO_pow, DirichletCharacter.norm_LFunction_product_ge_one, ValueDistribution.characteristic_pow_top, I_pow_three, cfcₙHom_real_eq_restrict, equivRealProdLm_symm_apply_re, UnitAddTorus.mFourierSubalgebra_separatesPoints, NumberField.mixedEmbedding.normAtPlace_neg, coe_algebraMap, Real.fourierIntegral_convergent_iff, NumberField.canonicalEmbedding.mem_rat_span_latticeBasis, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, PeriodPair.weierstrassPSeries_hasSum, AddChar.starComp_eq_inv, ofReal_comp_pow, NumberField.mixedEmbedding.unitSMul_smul, hasDerivAt_sin, fourierCoeff_eq_intervalIntegral, NumberField.mixedEmbedding.continuous_normAtPlace, exp_bound, taylorSeries_eq_on_eball, WStarAlgebra.exists_predual, span_fourierLp_closure_eq_top, LindemannWeierstrass.integral_exp_mul_eval, ModularFormClass.hasSum_qExpansion_of_abs_lt, LSeries.iteratedDeriv_alternating, GaussianInt.intCast_real_norm, WeakFEPair.functional_equation₀, NormedRing.algEquivComplexOfComplete_apply, Real.tendsto_integral_cexp_sq_smul, inner_map_self_eq_zero, hasStrictDerivAt_exp, circleAverage_sub_sub_inv_smul_of_differentiable_on, NumberField.mixedEmbedding.indexEquiv_apply_isReal, NumberField.ComplexEmbedding.Extension.comp_eq, Polynomial.mahlerMeasure_one, circleIntegral.integral_smul_const, hasStrictDerivAt_cosh, CStarAlgebra.span_nonneg_inter_closedBall, normSq_nonneg, wedgeIntegral_add_wedgeIntegral_eq, HurwitzZeta.sinZeta_two_mul_nat_add_one, GaussianInt.toComplex_im, GaussianInt.im_toComplex, NumberField.ComplexEmbedding.IsReal.comp, integral_boundary_rect_eq_zero_of_differentiable_on_off_countable, tsum_dirichletSummand, NumberField.InfinitePlace.Completion.bijective_extensionEmbedding_of_isComplex, NumberField.mixedEmbedding.normAtPlace_apply, EulerSine.tendsto_integral_cos_pow_mul_div, coe_selfAdjointEquiv, NumberField.ComplexEmbedding.lift_comp_algebraMap, Polynomial.fourierCoeff_toAddCircle_natCast, Polynomial.mahlerMeasure_zero, imaginaryPart_ofReal, ModularGroup.im_smul_eq_div_normSq, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, finrank_real_of_complex, CliffordAlgebraComplex.equiv_apply, ZMod.completedLFunction_one_sub_odd, NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt', NumberField.canonicalEmbedding.norm_le_iff, Real.fourier_bilin_convolution_eq_integral, coe_smul, NumberField.mixedEmbedding.normAtPlace_apply_of_isComplex, lift_symm_apply_coe, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, NumberField.mixedEmbedding_injective, NumberField.mixedEmbedding.forall_normAtPlace_eq_zero_iff, Real.tendsto_integral_exp_smul_cocompact, mellin_convergent_iff_norm, normSq_add_mul_I, Orientation.kahler_mul, GaussianFourier.integral_cexp_neg_mul_sum_add, ModularFormClass.qExpansion_coeff_eq_circleIntegral, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, NormedAlgebra.Real.nonempty_algEquiv_or, sin_eq_two_mul_tan_half_div_one_add_tan_half_sq, fourierIntegral_gaussian, EisensteinSeries.tendsto_double_sum_S_act, sinh_eq_tsum, Polynomial.isOpenQuotientMap_eval, GaussianInt.toComplex_neg, TemperedDistribution.instFourierInvSMul, PositiveLinearMap.ofPreGNS_toPreGNS, WeakFEPair.hf_modif_FE, integral_gaussian_sq_complex, ContinuousLinearMap.isPositive_iff_complex, DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, fourierIntegral_half_period_translate, NumberField.mixedEmbedding.logMap_apply, taylorSeries_eq_on_emetric_ball, TemperedDistribution.instFourierInvAdd, Polynomial.C_eq_or_isOpenQuotientMap_eval, Polynomial.leading_coeff_le_mahlerMeasure, NumberField.mixedEmbedding.latticeBasis_apply, selfAdjointEquiv_symm_apply, TemperedDistribution.fourierInv_apply, GaussianInt.re_toComplex, ofRealHom_eq_coe, sin_eq_tsum, ArithmeticFunction.LSeries_mul, cpow_nat_mul', NumberField.mixedEmbedding.fundamentalCone.intNorm_coe, NumberField.mixedEmbedding.norm_eq_zero_iff, NumberField.InfinitePlace.Completion.extensionEmbedding_of_isReal_coe, TemperedDistribution.fourierTransform_toTemperedDistributionCLM_eq, NumberField.norm_embedding_le_house, TemperedDistribution.instLineDerivAdd, Polynomial.supNorm_le_choose_natDegree_div_two_mul_mahlerMeasure, finrank_real_complex, NumberField.canonicalEmbedding_injective, hasStrictDerivAt_cpow_const, NumberField.InfinitePlace.LiesOver.mk_embedding_comp, NumberField.mixedEmbedding.normAtPlace_nonneg, ZMod.dft_const_mul, hasSum_taylorSeries_neg_log, VectorFourier.integral_bilin_fourierIntegral_eq_flip, unitary_iff_isStarNormal_and_spectrum_subset_unitary, finrank_real_complex_fact, NumberField.mixedEmbedding.commMap_apply_of_isReal, AlgHom.map_coe_real_complex, cexp_neg_quadratic_isLittleO_rpow_atTop, taylorSeries_eq_on_ball, TemperedDistribution.instLineDerivSMulComplex, ProbabilityTheory.integrable_pow_abs_mul_cexp_of_re_mem_interior_integrableExpSet, iteratedDeriv_odd_cos, exp_bound', ProbabilityTheory.integrable_pow_mul_cexp_of_re_mem_interior_integrableExpSet, Polynomial.logMahlerMeasure_zero, WeakFEPair.functional_equation, I_sq, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add, sq_nonpos_iff, cpow_nat_inv_pow, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, NumberField.mixedEmbedding.latticeBasis_repr_apply, ModularFormClass.qExpansion_isBigO, NumberField.canonicalEmbedding.apply_at, TemperedDistribution.laplacian_toTemperedDistributionCLM_eq, ProbabilityTheory.complexMGF_id_gaussianReal, mul_conj, exp_rat_mul_pi_mul_I_pow_two_mul_den, VonNeumannAlgebra.coe_toStarSubalgebra, rank_real_of_complex, qExpansion_mul, ValueDistribution.proximity_pow_top, NumberField.AdeleRing.algebraMap_fst_apply, VectorFourier.fourierPowSMulRight_apply, skewAdjointPart_eq_I_smul_imaginaryPart, inv_im, NumberField.ComplexEmbedding.IsUnmixed.isReal_iff_isReal, WithCStarModule.inner_single_left, NumberField.mixedEmbedding.normAtPlace_negAt, EisensteinSeries.G2_eq_tsum_cexp, Circle.normSq_coe, PositiveLinearMap.leftMulMapPreGNS_mul_eq_comp, hasDerivAt_Gamma_one, NumberField.mixedEmbedding.normAtPlace_smul, PeriodPair.hasSum_weierstrassPExcept, GammaSeq_mul, ProbabilityTheory.hasDerivAt_iteratedDeriv_complexMGF, qExpansion_mul_coeff_zero, SchwartzMap.integral_fourier_smul_eq, GaussianInt.toComplex_zero, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, EisensteinSeries.qExpansion_identity, iteratedDeriv_even_sin, fourierSubalgebra_closure_eq_top, GaussianFourier.integral_cexp_neg_sum_mul_add, NumberField.mixedEmbedding.fractionalIdealLatticeBasis_apply, Unitization.complex_cfcₙ_eq_cfc_inr, tendsto_one_add_pow_exp_of_tendsto, Real.nonempty_algEquiv_or, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, log_sub_logTaylor_isBigO, AddChar.doubleDualEquiv_symm_doubleDualEmb_apply, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, toMatrix_conjAe, inner_map_polarization, ModularFormClass.hasSum_qExpansion_of_norm_lt, TemperedDistribution.derivCLM_toTemperedDistributionCLM_eq, NumberField.InfinitePlace.embedding_injective, ArithmeticFunction.LSeriesSummable_mul, jacobiTheta_eq_tsum_nat, PositiveLinearMap.norm_apply_le_of_nonneg, NumberField.InfinitePlace.Completion.isometry_extensionEmbedding_of_isReal, taylorSeries_eq_on_eball', VonNeumannAlgebra.mem_commutant_iff, hasDerivAt_Gamma_one_half, normSq_eq_conj_mul_self, ModularForm.slash_def, ZMod.invDFT_def, NumberField.InfinitePlace.mkComplex_coe, MeasureTheory.Lp.ker_toTemperedDistributionCLM_eq_bot, MeasureTheory.Lp.toTemperedDistribution_toLp_eq, Circle.coeHom_apply, Real.tendsto_integral_gaussian_smul, VectorFourier.fourierIntegral_comp_add_right, UnitAddTorus.span_mFourier_closure_eq_top, TemperedDistribution.laplacian_apply_apply, TemperedDistribution.smulLeftCLM_smulLeftCLM_apply, tendsto_integral_exp_inner_smul_cocompact, imaginaryPart_smul, CliffordAlgebraComplex.toComplex_involute, NumberField.mixedEmbedding.norm_real, VectorFourier.fourierIntegral_probChar, Polynomial.logMahlerMeasure_const, PeriodPair.weierstrassPExceptSeries_hasSum, arg_pow_coe_angle, Fourier.fourierIntegral_def, CliffordAlgebraComplex.toComplex_ofComplex, isOpenQuotientMap_pow_compl_zero, TemperedDistribution.lineDerivOp_apply_apply, rank_rat_complex, Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv, exp_nat_mul, TemperedDistribution.lineDerivOp_fourierInv_eq, cos_sq, hasSum_taylorSeries_log, fourierCoeffOn_eq_integral, re_expect, VonNeumannAlgebra.coe_commutant, PeriodPair.hasSum_sumInvPow, hasDerivAt_tan, CompletelyPositiveMap.map_cstarMatrix_nonneg', integral_gaussian_complex, circleMap_zero_pow, tan_sq_div_one_add_tan_sq, summable_bernoulli_fourier, NumberField.mixedEmbedding.indexEquiv_apply_isComplex_snd, Polynomial.logMahlerMeasure_monomial, WeakFEPair.f_modif_aux1, NumberField.InfinitePlace.eq_iff_eq, GaussianFourier.integrable_cexp_neg_sum_mul_add, hasSum_sinh, norm_exp_sub_sum_le_exp_norm_sub_sum, Polynomial.logMahlerMeasure_C_mul_X_add_C, MeasureTheory.Lp.toTemperedDistribution_apply, TemperedDistribution.smulLeftCLM_const, iteratedDeriv_cexp_const_mul, Real.iteratedDeriv_fourierIntegral, adjoin_I, TemperedDistribution.smulLeftCLM_sum, NumberField.InfinitePlace.Completion.WithAbs.ratCast_equiv, hasSum_sin', jacobiTheta₂_fderiv_undef, DirichletCharacter.convolution_mul_moebius, Quaternion.coe_ofComplex, UpperHalfPlane.σ_conj, NumberField.mixedEmbedding.fundamentalCone.realSpaceToLogSpace_expMap_symm, hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, TemperedDistribution.instContinuousFourierInv, sin_eq_tsum', VonNeumannAlgebra.instStarMemClass, NormedAlgebra.Complex.nonempty_algEquiv, normSq_apply, NumberField.mixedEmbedding.fundamentalCone.mem_idealSet, TemperedDistribution.fourier_apply, NumberField.prod_archAbsVal_eq, NumberField.ComplexEmbedding.isReal_comp_iff, RCLike.normSq_to_complex, MeasureTheory.Integrable.fourier_smul, NumberField.ComplexEmbedding.IsConj.comp, ZMod.completedLFunction_one_sub_even, VectorFourier.integral_fourierIntegral_smul_eq_flip, BoundedContinuousFunction.charMonoidHom_apply, Polynomial.cyclotomic_mahlerMeasure_eq_one, sin_sq, SchwartzMap.coe_apply, integral_mul_cpow_one_add_sq, im_sq_le_normSq, IsExpCmpFilter.isLittleO_pow_mul_exp, CliffordAlgebraComplex.ofComplex_comp_toComplex, NumberField.mixedEmbedding.fundamentalCone.sum_expMap_symm_apply, realPart_smul, integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn, AddChar.doubleDualEmb_doubleDualEquiv_symm_apply, AddChar.coe_complexBasis, Polynomial.toAddCircle_X_eq_fourier_one, DirichletCharacter.isMultiplicative_zetaMul, ModularForm.qExpansion_mul, CircleIntegrable.out, VonNeumannAlgebra.instSubringClass, LSeries_hasDerivAt, CStarAlgebra.span_nonneg_inter_ball, Real.hasDerivAt_fourierIntegral, NumberField.mixedEmbedding.mixedEmbedding_apply_isComplex, NumberField.canonicalEmbedding_eq_basisMatrix_mulVec, integrable_cexp_quadratic, NumberField.mixedEmbedding.fundamentalCone.mem_integerSet, TemperedDistribution.smulLeftCLM_smul, UpperHalfPlane.denom_cocycle', ofRealAm_coe, ArithmeticFunction.iteratedDeriv_LSeries_alternating, two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, Circle.toUnits_apply, fourierCoeffOn.const_smul, Real.fourierInv_eq, inv_one_add_tan_sq, isConformalMap_complex_linear, cpow_mul_ofNat, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, isAbsoluteValueNorm, hasSum_arctan, differentiableWithinAt_complex_iff_differentiableWithinAt_real, TemperedDistribution.derivCLM_apply_apply, hasDerivAt_exp, TorusIntegrable.function_integrable, DirichletCharacter.fourierTransform_eq_gaussSum_mulShift, hasSum_cosh, riemannZeta_neg_nat_eq_bernoulli, NumberField.mixedEmbedding.disjoint_span_commMap_ker, taylorSeries_eq_of_entire', UpperHalfPlane.σ_mul_comm, NumberField.ComplexEmbedding.conjugate_coe_eq, PeriodPair.summable_weierstrassPExceptSummand, NumberField.ComplexEmbedding.lift_algebraMap_apply, integrable_mul_cexp_neg_mul_sq, NumberField.InfiniteAdeleRing.ringEquiv_mixedSpace_apply, hasDerivAt_log_sub_logTaylor, Polynomial.norm_coeff_le_choose_mul_mahlerMeasure, ZMod.dft_eq_fourier, ZMod.dft_odd_iff, AddChar.complexBasis_apply, riemannZeta_two_mul_nat, ofReal_pow, Polynomial.finite_mahlerMeasure_le, skewAdjoint.I_smul_neg_I, coe_basisOneI, ProbabilityTheory.charFun_gaussianReal, hasDerivAt_log, TemperedDistribution.fourierTransformInv_apply, hasSum_cos, logTaylor_succ, LinearMap.isSymmetric_iff_inner_map_self_real, ModularFormClass.qExpansionFormalMultilinearSeries_coeff, eqOn_iteratedDerivWithin_cotTerm_integerComplement, zeta_nat_eq_tsum_of_gt_one, spectrum.map_polynomial_aeval, Real.fourierInv_eq', affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div', MellinConvergent.cpow_smul, Real.fourierIntegral_deriv, NumberField.InfinitePlace.Completion.Rat.norm_infinitePlace_completion, AddChar.expect_apply_eq_ite, Polynomial.toAddCircle_C_eq_smul_fourier_zero, isCauSeq_exp, VectorFourier.hasFDerivAt_fourierChar_smul, BoundedContinuousFunction.separatesPoints_charPoly, integrable_cexp_quadratic', one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, Real.fourierIntegral_real_eq, Polynomial.Gal.splits_ℚ_ℂ, circleAverage_sub_sub_inv_smul_of_differentiable_on_off_countable, ZMod.injective_stdAddChar, qExpansion_of_pow, UnitAddTorus.span_mFourierLp_closure_eq_top, cpow_ofNat_inv_pow, ZMod.LFunction_dft, qExpansion_smul, WeakFEPair.f_modif_aux2, AddChar.zmodAddEquiv_apply, NumberField.InfinitePlace.card_complex_embeddings, mellin_cpow_smul, ZMod.dft_const_smul, cos_sq', NumberField.Units.complexEmbedding_inj, normSq_neg, NumberField.mixedEmbedding.commMap_canonical_eq_mixed, Polynomial.fourierCoeff_toAddCircle, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, ZMod.dft_smul_const, NumberField.mixedEmbedding.norm_negAt, NumberField.mixedEmbedding.norm_unit, summable_pow_mul_cexp, cpow_ofNat, WeakFEPair.Λ_residue_zero, ModularFormClass.hasSum_qExpansion, qExpansion_coeff_isBigO_of_norm_isBigO, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, Real.fourierIntegralInv_eq, WithCStarModule.prod_inner, tendsto_tsum_powerSeries_nhdsWithin_stolzSet, cpow_two, summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp, UpperHalfPlane.c_mul_im_sq_le_normSq_denom, isConformalMap_iff_is_complex_or_conj_linear, ZMod.dft_def, range_normSq, NumberField.InfinitePlace.apply, Polynomial.mahlerMeasure_le_sum_norm_coeff, Real.fourier_eq', UpperHalfPlane.smulAux'_im, WeakFEPair.Λ_residue_k, integrable_cexp_neg_mul_sq, sq_cpow_two_inv, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, Fourier.fourierIntegral_comp_add_right, affine_of_mapsTo_ball_of_norm_dslope_eq_div, NumberField.mixedEmbedding.normAtPlace_mixedSpaceOfRealSpace, imLm_coe, normSq_ofReal_sub_I_mul_sqrt_one_sub, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, PeriodPair.derivWeierstrassPExcept_def, NumberField.mixedEmbedding.nnnorm_eq_sup_normAtPlace, GaussianInt.toComplex_one, EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act, NumberField.InfinitePlace.comap_embedding_of_isReal, LSeries.hasDerivAt_term, normSq_eq_zero, CStarAlgebra.span_unitary, fourier_gaussian_pi, Real.fourier_smul_convolution_eq, UpperHalfPlane.σ_denom, NumberField.house.basis_repr_norm_le_const_mul_house, TemperedDistribution.laplacianCLM_apply, continuousAt_gaussian_integral, NumberField.mixedEmbedding.fundamentalCone.expMap_basis_of_ne, Real.deriv_fourier, CStarAlgebra.exists_sum_four_nonneg, taylorSeries_eq_on_emetric_ball', GaussianInt.normSq_div_sub_div_lt_one, ZMod.stdAddChar_apply, PositiveLinearMap.preGNS_norm_sq, EulerSine.integral_cos_mul_cos_pow_even, Polynomial.card_mahlerMeasure_le_prod, MeasureTheory.Lp.toTemperedDistribution_smul_eq, NumberField.house_eq_sup', GaussianInt.intCast_complex_norm, nonempty_linearEquiv_real, LSeries_convolution, GaussianInt.intCast_re, ofReal_choose, NumberField.mixedEmbedding.convexBodySum_mem, one_div_sub_hasFPowerSeriesOnBall_zero, ModularGroup.smul_eq_lcRow0_add, fourierIntegral_gaussian_pi', tendsto_tsum_powerSeries_nhdsWithin_lt, rank_real_complex, CStarMatrix.norm_def, tan_two_mul, integral_gaussian_complex_Ioi, TemperedDistribution.smulLeftCLM_apply_apply, conjCLE_coe, NumberField.mixedEmbedding.mem_rat_span_latticeBasis, Circle.norm_smul, NumberField.mixedEmbedding.norm_apply, mellin_comp_mul_left, GaussianInt.toComplex_div_im, BoundedContinuousFunction.char_mem_charPoly, NumberField.mixedEmbedding.fundamentalCone.normAtAllPlaces_normLeOne, ZMod.dft_apply_zero, differentiableAt_iteratedDerivWithin_cexp, pi_mul_cot_pi_q_exp, PeriodPair.weierstrassPExcept_add, hasDerivAt_Gamma_nat, CStarMatrix.inner_toCLM_conjTranspose_right, norm_jacobiTheta₂_term_fderiv_le, tan_eq_one_sub_tan_half_sq_div_one_add_tan_half_sq, GaussianInt.toComplex_eq_zero, normSq_one, mellin_comp_mul_right, NonUnitalStarAlgHom.instContinuousLinearMapClassComplex, NumberField.mixedEmbedding.normAtPlace_real, CStarMatrix.toCLM_apply_single_apply, map_isSelfAdjoint, ModularForm.slash_apply, NumberField.mixedEmbedding.fundamentalCone.intNorm_idealSetEquiv_apply, restrictScalars_toSpanSingleton', hasSum_taylorSeries_on_eball, Polynomial.fourierCoeff_toAddCircle_eq_zero_of_lt_zero, UpperHalfPlane.petersson_slash, Polynomial.int_cyclotomic_spec
instStarRing 📖	CompOp	190 math math: norm_conj, log_conj_eq_ite, Orientation.kahler_map_complex, jacobiTheta₂_conj, IsStarNormal.instNonUnitalContinuousFunctionalCalculus, sinh_conj, coe_realPart, SchwartzMap.integral_sesq_fourier_fourier, lim_conj, AlgHomClass.instStarHomClass, NonUnitalCStarAlgebra.toStarModule, kahler, areaForm, re_eq_add_conj, CliffordAlgebraComplex.ofComplex_conj, CStarModule.innerₛₗ_apply, realPart_ofReal, conj_inv, BoundedContinuousFunction.char_neg, Unitization.real_cfcₙ_eq_cfc_inr, NumberField.mem_maximalRealSubfield_iff, star_def, nndist_self_conj, MeasureTheory.charFun_neg, StarAlgebra.elemental.characterSpaceToSpectrum_coe, Circle.ofConjDivSelf_coe, conjLIE_apply, Circle.starRingEnd_addChar, cosh_conj, oangle, UnitAddTorus.mFourierSubalgebra_coe, CStarModule.innerSL_apply, UnitAddTorus.mFourierSubalgebra_closure_eq_top, gelfandStarTransform_naturality, isCauSeq_conj, gelfandStarTransform_apply_apply, jacobiTheta₂'_conj, cos_conj, conj_rootsOfUnity, Unitization.instStarOrderedRing, instContinuousStar, SchwartzMap.integral_sesq_fourier_eq, isometry_conj, summable_conj, log_inv_eq_ite, mul_conj', Orientation.kahler_swap, UnitDisc.coe_conj, BoundedContinuousFunction.mem_charPoly, Gamma_conj, NumberField.ComplexEmbedding.IsConj.eq, ringHom_eq_id_or_conj_of_continuous, dist_self_conj, gelfandStarTransform_symm_apply, fourier_neg', CStarModule.inner_smul_left_complex, normSq_add, UpperHalfPlane.sigma_J, VonNeumannAlgebra.centralizer_centralizer', inv_cpow_eq_ite', StarAlgebra.elemental.bijective_characterSpaceToSpectrum, gelfandTransform_map_star, cfcₙ_real_eq_complex, UnitAddTorus.mFourier_neg, normSq_sub, Circle.coe_inv_eq_conj, sub_conj, Orientation.kahler_neg_orientation, IsSelfAdjoint.cfc_arg, conj_im, fourierSubalgebra_separatesPoints, continuous_conj, conj_cpow, VonNeumannAlgebra.mem_carrier, selfAdjointEquiv_apply, NumberField.IsCMField.complexEmbedding_complexConj, inv_cpow_eq_ite, StarSubalgebra.spectrum_eq, inv_def, cfcHom_real_eq_restrict, arg_conj, inner_map_complex, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, AnalyticAt.harmonicAt_conj, GaussianInt.toComplex_star, Circle.star_addChar, conjAe_coe, conj_mul', inr_comp_cfcₙHom_eq_cfcₙAux, sin_conj, conj_eq_iff_re, CommCStarAlgebra.toStarModule, conj_cpow_eq_ite, normSq_conj, add_conj, tanh_conj, UpperHalfPlane.J_smul, spectrum_subset_unitary_of_mem_unitary, AddChar.starComp_apply, cot_conj, SchwartzMap.integral_sesq_fourierIntegral_eq, NonUnitalCommCStarAlgebra.toStarModule, UpperHalfPlane.petersson_symm, instStarModuleReal, fourierSubalgebra_coe, inner, MulChar.star_apply', UpperHalfPlane.coe_J_smul, NumberField.conj_basisMatrix, Unitization.nnreal_cfcₙ_eq_cfc_inr, cfcₙHom_real_eq_restrict, UnitAddTorus.mFourierSubalgebra_separatesPoints, AddChar.starComp_eq_inv, WStarAlgebra.exists_predual, Orientation.kahler_rotation_left, hasSum_conj, IsStarNormal.instContinuousFunctionalCalculus, coe_selfAdjointEquiv, dist_conj_conj, conjCLE_apply, arg_conj_coe_angle, nndist_conj_self, imaginaryPart_ofReal, UpperHalfPlane.cosh_half_dist, StarSubalgebra.mem_spectrum_iff, MulChar.star_eq_inv, UpperHalfPlane.exp_half_dist, uniformContinuous_ringHom_eq_id_or_conj, fourier_neg, conj_re, im_eq_sub_conj, conj_ofNat, UnitAddTorus.hasSum_prod_mFourierCoeff, BoundedContinuousFunction.star_mem_range_charAlgHom, selfAdjointEquiv_symm_apply, hasSum_conj', exp_conj, unitary_iff_isStarNormal_and_spectrum_subset_unitary, UpperHalfPlane.sinh_half_dist_add_dist, LinearIsometry.im_apply_eq_im, mul_conj, IsStarNormal.instNonUnitalIsometricContinuousFunctionalCalculus, nndist_conj_comm, VonNeumannAlgebra.coe_toStarSubalgebra, StarSubalgebra.coe_isUnit, conj_natCast, nndist_conj_conj, instContinuousMapUniqueHom, Orientation.areaForm_map_complex, cpow_conj, fourierSubalgebra_closure_eq_top, Unitization.complex_cfcₙ_eq_cfc_inr, conj_eq_iff_real, normSq_eq_conj_mul_self, conj_eq_iff_im, dist_conj_comm, GammaIntegral_conj, WeakDual.Complex.instStarHomClass, CliffordAlgebraComplex.toComplex_involute, cfc_real_eq_complex, conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj, HurwitzZeta.jacobiTheta₂''_conj, UpperHalfPlane.σ_conj, NumberField.canonicalEmbedding.conj_apply, im_eq_zero_iff_isSelfAdjoint, conj_ofReal, Orientation.oangle_map_complex, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, inv_eq_conj, conj_neg_I, CStarAlgebra.toStarModule, UpperHalfPlane.tanh_half_dist, Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero, conj_tsum, NumberField.ComplexEmbedding.conjugate_coe_eq, IsStarNormal.instIsometricContinuousFunctionalCalculus, Unitary.argSelfAdjoint_coe, tan_conj, LinearMap.isSymmetric_iff_inner_map_self_real, BoundedContinuousFunction.separatesPoints_charPoly, UnitDisc.coe_star, cfcHom_eq_of_isStarNormal, log_conj, conj_I, continuousFunctionalCalculus_map_id, CFC.complex_exp_eq_normedSpace_exp, dist_conj_self, WeakDual.CharacterSpace.instStarHomClass, BoundedContinuousFunction.char_mem_charPoly, Unitization.cfcₙ_eq_cfc_inr
instSub 📖	CompOp	439 math math: HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip_eps, cos_sub_pi, norm_log_one_sub_inv_add_logTaylor_neg_le, HurwitzZeta.differentiableAt_hurwitzZetaEven_sub_one_div, EulerSine.integral_cos_mul_cos_pow, tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero, ProbabilityTheory.isGaussian_iff_gaussian_charFun, HurwitzZeta.differentiableAt_hurwitzZeta_sub_one_div, cos_three_mul, EisensteinSeries.hasSum_e2Summand_symmetricIcc, sin_pi_div_two_sub, PeriodPair.hasSumLocallyUniformly_weierstrassP, circleTransformDeriv_eq, HurwitzZeta.hasSum_int_cosKernel₀, norm_sub_one_sq_eq_of_norm_eq_one, norm_sub_eq, tan_add_mul_I, cos_sub_sin_I, tan_nat_mul_pi_sub, DirichletCharacter.LSeries_eulerProduct_tprod, norm_sub_mem_Icc_angle, PeriodPair.derivWeierstrassPExcept_sub, ProbabilityTheory.hasGaussianLaw_iff_charFunDual_map_eq, countingFunction_finsum_eq_finsum_add, HurwitzZeta.sinZeta_neg_two_mul_nat_sub_one, ZMod.LFunction_one_sub, hasSum_taylorSeries_of_entire, deriv_cpow_const, tan_pi_sub, circleMap_neg_pi_div_two, affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div, circleIntegral.integral_sub_inv_of_mem_ball, betaIntegral_scaled, cos_sub, WeakFEPair.symm_Λ₀_eq, circleAverage_log_norm_sub_const₁, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq, ProbabilityTheory.hasGaussianLaw_iff_charFun_map_eq, hasSum_taylorSeries_on_ball, cosh_three_mul, HasStrictDerivAt.cpow, sum_cauchyPowerSeries_eq_integral, tan_pi_div_two_sub, Real.norm_exp_I_mul_ofReal_sub_one_le, taylorSeries_eq_of_entire, circleAverage_log_norm_sub_const_of_mem_closedBall, hasSum_nat_jacobiTheta, PeriodPair.hasSum_derivWeierstrassPExcept, digamma_one_half, DifferentiableOn.circleIntegral_sub_inv_smul, HarmonicAt.analyticAt_complex_partial, Real.nnnorm_exp_I_mul_ofReal_sub_one_le, ModularForm.tsum_logDeriv_eta_q, EisensteinSeries.summable_left_one_div_linear_sub_one_div_linear, mul_angle_le_norm_sub, integral_circleTransform, tan_eq, HasDerivAt.cpow, DirichletCharacter.LFunction_changeLevel, cos_eq_two_mul_tan_half_div_one_sub_tan_half_sq, DirichletCharacter.IsPrimitive.completedLFunction_one_sub, circleMap_sub_center, EisensteinSeries.E2_slash_action, sin_int_mul_two_pi_sub, sub_im, cos_nat_mul_two_pi_sub_pi, sin_bound, hasDerivAt_GammaIntegral, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, deriv_ofReal_cpow_const, Function.Periodic.cuspFunction_sub, two_sinh, PeriodPair.ite_eq_one_sub_sq_mul_weierstrassP, EisensteinSeries.G2_slash_action, taylorSeries_eq_on_ball', StrongFEPair.functional_equation, circleAverage_log_norm_sub_const_eq_log_radius_add_posLog, circleIntegral.integral_sub_inv_smul_sub_smul, LindemannWeierstrass.exp_polynomial_approx, hasStrictFDerivAt_cpow', cos_eq_iff_quadratic, riemannZeta_one_sub, PeriodPair.hasSum_derivWeierstrassP, PeriodPair.weierstrassP_bound, HurwitzZeta.differentiable_hurwitzZeta_sub_hurwitzZeta, PeriodPair.hasSumLocallyUniformly_derivWeierstrassP, exp_eq_exp_iff_exp_sub_eq_one, NumberField.tendsto_sub_one_mul_dedekindZeta_nhdsGT, cauchyPowerSeries_apply, sin_sub_nat_mul_two_pi, cosh_sq_sub_sinh_sq, DiffContOnCl.circleIntegral_sub_inv_smul, sin_nat_mul_two_pi_sub, HurwitzZeta.expZeta_one_sub, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, LSeries_sub, Real.circleAverage_eq_circleIntegral, norm_exp_I_mul_ofReal_sub_one, integral_exp_mul_complex, ofReal_comp_sub, sub_re, cot_series_rep', cos_pi_sub, LSeries.term_sub, norm_log_sub_logTaylor_le, HadamardThreeLines.mem_verticalClosedStrip_of_scale_id_mem_verticalClosedStrip, cpow_sub, cos_two_mul, tendsto_sub_mul_tsum_nat_cpow, cos_sub_two_pi, partialGamma_add_one, sin_sub_two_pi, sin_sub_pi_div_two, DirichletCharacter.LSeries_changeLevel, integral_cpow, norm_log_one_add_sub_self_le, EisensteinSeries.tendsto_zero_inv_linear_sub, hasStrictFDerivAt_cpow, tendsto_riemannZeta_sub_one_div, ZetaAsymptotics.tendsto_Gamma_term_aux, inner_map_polarization', ZMod.LFunction_neg_two_mul_nat_sub_one, HasFDerivWithinAt.cpow, EisensteinSeries.G2_S_transform, SlashInvariantForm.sub_apply, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, riemannZeta_eulerProduct_exp_log, cos_int_mul_two_pi_sub, HarmonicAt.differentiableAt_complex_partial, ProbabilityTheory.IsGaussian.charFun_eq', logDeriv_sin_div_eq_cot, HurwitzZeta.cosZeta_one_sub, EulerSine.sin_pi_mul_eq, isBigO_re_sub_re, circleAverage_log_norm_sub_const₀, WeakFEPair.Λ₀_eq, log_exp_eq_sub_toIocDiv, circleIntegral.integral_sub_zpow_of_undef, norm_exp_sub_sum_le_norm_mul_exp, tan_sub', DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, HurwitzZeta.completedHurwitzZetaEven_residue_one, sinh_sq, HadamardThreeLines.interpStrip_scale, normSq_sub, tan_add, PeriodPair.weierstrassPExcept_def, circleIntegrable_log_norm_factorizedRational, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, sub_conj, EisensteinSeries.hasSum_e2Summand_symmetricIco, HasDerivWithinAt.cpow, norm_jacobiTheta_sub_one_le, tan_add', TemperedDistribution.smulLeftCLM_sub, DirichletCharacter.LFunctionTrivChar_eq_mul_riemannZeta, hasSum_cauchyPowerSeries_integral, tan_sub_int_mul_pi, HurwitzZeta.completedHurwitzZetaEven_eq, sin_eq_sin_iff, isBigO_riemannZeta_sub_one_div, hasFDerivAt_cpow, ProbabilityTheory.isGaussian_iff_charFun_eq, sin_pi_sub, ModularForm.differentiableAt_eta_tprod, HadamardThreeLines.interpStrip_eq_of_pos, sin_three_mul, eqOn_iteratedDerivWithin_cotTerm_upperHalfPlaneSet, ModularFormClass.exp_decay_sub_atImInfty, one_div_sub_sq_hasFPowerSeriesOnBall_zero, DirichletCharacter.summable_neg_log_one_sub_mul_prime_cpow, ModularForm.sub_apply, PeriodPair.hasSumLocallyUniformly_weierstrassPExcept, completedRiemannZeta_eq, log_sub_self_isBigO, ModularForm.logDeriv_one_sub_mul_cexp_comp, two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, PeriodPair.derivWeierstrassP_sq, cos_sub_nat_mul_two_pi, hasSum_two_pi_I_cauchyPowerSeries_integral, eqOn_iteratedDeriv_cotTerm, SlashInvariantForm.coe_sub, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, GaussianInt.toComplex_sub, cos_two_mul', tendsto_euler_sin_prod, ZMod.completedLFunction_eq, HurwitzZeta.completedSinZeta_one_sub, tan_sub, EulerSine.integral_sin_mul_sin_mul_cos_pow_eq, HurwitzZeta.hurwitzZeta_residue_one, isBigO_at_im_infty_jacobiTheta_sub_one, ProbabilityTheory.IsGaussian.charFunDual_eq, PeriodPair.derivWeierstrassP_sub_coe, cot_pi_eq_exp_ratio, DiffContOnCl.circleAverage_smul_div, HurwitzZeta.differentiableAt_one_completedHurwitzZetaEven_sub_completedHurwitzZetaEven, HurwitzZeta.hurwitzZetaEven_one_sub, LSeries.term_sub_apply, HurwitzZeta.completedCosZeta₀_one_sub, qExpansion_sub, cot_series_rep, Summable.clog_one_sub, HurwitzZeta.completedHurwitzZetaEven_one_sub, cos_add, tan_sub_nat_mul_pi, abel_aux, norm_sub_le_angle, cosh_sub, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, integral_cexp_quadratic, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, tan_sub_pi, cos_bound, dist_eq, HasStrictDerivAt.cpow_const, CuspForm.coe_sub, one_div_sub_pow_hasFPowerSeriesOnBall_zero, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, ModularForm.logDeriv_one_sub_cexp, hasSum_taylorSeries_on_emetric_ball, hasFPowerSeriesOn_cauchy_integral, circleIntegral_div_sub_of_differentiable_on_off_countable, circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable, PeriodPair.weierstrassP_sub_coe, ProbabilityTheory.HasGaussianLaw.charFun_map_eq, isBigO_im_sub_im, angle_le_mul_norm_sub, PeriodPair.hasSum_weierstrassP, cos_add_cos, LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_and_nonneg, jacobiTheta₂'_add_left', jacobiTheta₂'_functional_equation, riemannZeta_one, exp_sub_sum_range_succ_isLittleO_pow, continuousOn_prod_circle_transform_function, MeasureTheory.measureReal_abs_gt_le_integral_charFun, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, cos_sub_pi_div_two, hasSum_arctan_aux, MeromorphicOn.circleAverage_log_norm, DirichletCharacter.deriv_LFunctionTrivChar₁_apply_of_ne_one, tan_int_mul_pi_sub, tendsto_logDeriv_euler_cot_sub, cos_add_mul_I, exp_sub_sum_range_isBigO_pow, riemannZeta_eulerProduct, EisensteinSeries.summable_linear_sub_mul_linear_add, MeasureTheory.measureReal_abs_dual_gt_le_integral_charFunDual, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, exp_sub_cosh, betaIntegral_convergent, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, cos_int_mul_two_pi_sub_pi, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, ModularForm.summable_logDeriv_one_sub_eta_q, exp_bound, taylorSeries_eq_on_eball, MeasureTheory.norm_one_sub_charFun_le_two, circleIntegral.integral_sub_zpow_of_ne, DirichletCharacter.LSeries_eulerProduct_hasProd, LSeriesHasSum.sub, WeakFEPair.functional_equation₀, Real.enorm_exp_I_mul_ofReal_sub_one_le, circleAverage_sub_sub_inv_smul_of_differentiable_on, EisensteinSeries.q_expansion_bernoulli, cotTerm_identity, norm_sub_one_sq_eqOn_sphere, sqrt_neg_I, circleIntegrable_sub_zpow_iff, completedRiemannZeta₀_one, fourierCoeffOn_of_hasDerivAt_Ioo, HurwitzZeta.hurwitzZetaOdd_eq, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, DirichletCharacter.LSeries_eulerProduct_exp_log, ZMod.completedLFunction_one_sub_odd, HadamardThreeLines.interpStrip_eq_of_mem_verticalStrip, mellin_convergent_iff_norm, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat_sub_one, ProbabilityTheory.isGaussian_iff_charFunDual_eq, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, norm_exp_sub_one_le, circleAverage_log_norm_sub_const_eq_posLog, HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc, im_eq_sub_conj, HurwitzZeta.hurwitzZeta_one_sub, circleAverage_log_norm_factorizedRational, DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, taylorSeries_eq_on_emetric_ball, hasDerivAt_ofReal_cpow_const, LSeriesSummable.sub, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun, Gammaℝ_one_sub_mul_Gammaℝ_one_add, MeasureTheory.measureReal_abs_inner_gt_le_integral_charFun, exp_sub_pi_mul_I, norm_sub_eq_iff, hasStrictDerivAt_cpow_const, GammaSeq_eq_approx_Gamma_integral, hasSum_taylorSeries_neg_log, exp_sub_sinh, taylorSeries_eq_on_ball, exp_bound', sin_sub, WeakFEPair.functional_equation, circleAverage_log_norm_sub_const₂, cos_eq, sin_add_sin, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, HasStrictFDerivAt.cpow, deriv_cpow_const, HurwitzZeta.completedCosZeta_eq, HurwitzZeta.completedCosZeta_one_sub, cuspFunction_sub, CuspForm.sub_apply, EisensteinSeries.G2_eq_tsum_cexp, PeriodPair.hasSum_weierstrassPExcept, HadamardThreeLines.eventuallyle, completedRiemannZeta₀_one_sub, GammaSeq_mul, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, sin_sub_pi, circleIntegrable_sub_inv_iff, LSeries.abscissaOfAbsConv_sub_le, cos_pi_div_two_sub, ofReal_sub, log_sub_logTaylor_isBigO, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, cos_sub_cos, ArithmeticFunction.LSeries_zeta_eulerProduct_exp_log, inner_map_polarization, taylorSeries_eq_on_eball', norm_log_one_sub_inv_sub_self_le, PeriodPair.derivWeierstrassPExcept_add_coe, sin_sub_int_mul_two_pi, bernoulliFourierCoeff_recurrence, exp_sub, Gamma_mul_Gamma_add_half, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, sin_two_pi_sub, cot_eq_exp_ratio, norm_exp_sub_one_sub_id_le, PeriodPair.hasSum_sumInvPow, DirichletCharacter.LFunctionTrivChar_residue_one, ProbabilityTheory.IsGaussian.charFun_eq, completedRiemannZeta_residue_one, circleIntegral.integral_sub_center_inv, norm_exp_sub_sum_le_exp_norm_sub_sum, completedRiemannZeta_one, DirichletCharacter.Odd.LFunction_neg_two_mul_nat_sub_one, Gammaℝ_div_Gammaℝ_one_sub, EisensteinSeries.summable_right_one_div_linear_sub_one_div_linear_succ, HasDerivWithinAt.cpow_const, Gamma_mul_Gamma_one_sub, circleIntegrable_log_norm_sub_const, betaIntegral_convergent_left, ZMod.LFunction_residue_one, riemannZeta_eulerProduct_tprod, ZMod.completedLFunction_one_sub_even, sinh_sub, sin_sq, integral_mul_cpow_one_add_sq, HadamardThreeLines.scale_id_mem_verticalStrip_of_mem_verticalStrip, riemannZeta_residue_one, one_div_one_sub_hasFPowerSeriesOnBall_zero, cos_nat_mul_two_pi_sub, fourierCoeffOn_of_hasDerivAt, HurwitzZeta.tendsto_hurwitzZeta_sub_one_div_nhds_one, two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, inv_Gammaℝ_one_sub, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, ModularForm.coe_sub, taylorSeries_eq_of_entire', HasFDerivAt.cpow, hasDerivAt_log_sub_logTaylor, GammaIntegral_convergent, HurwitzZeta.sinZeta_eq, cos_two_pi_sub, ProbabilityTheory.IsGaussian.charFunDual_eq', HurwitzZeta.completedHurwitzZetaOdd_one_sub, ProbabilityTheory.charFun_gaussianReal, sin_sub_sin, HurwitzZeta.hurwitzZetaOdd_one_sub, eqOn_iteratedDerivWithin_cotTerm_integerComplement, affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div', HurwitzZeta.hasSum_nat_hurwitzZetaOdd_of_mem_Icc, riemannZeta_eulerProduct_hasProd, torusMap_sub_center, DirichletCharacter.LSeries_eulerProduct, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, circleAverage_sub_sub_inv_smul_of_differentiable_on_off_countable, WeakFEPair.f_modif_aux2, cos_eq_cos_iff, cos_sq', integral_cpow_mul_exp_neg_mul_Ioi, norm_sub_one_sq_eq_of_norm_one, integral_cos_mul_complex, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, HurwitzZeta.sinZeta_one_sub, HurwitzZeta.completedHurwitzZetaEven₀_one_sub, approx_Gamma_integral_tendsto_Gamma_integral, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq, inv_Gammaℝ_two_sub, ModularFormClass.exp_decay_sub_atImInfty', WeakFEPair.Λ_residue_k, HadamardThreeLines.scale_id_mem_verticalClosedStrip_of_mem_verticalClosedStrip, fourierCoeffOn_of_hasDeriv_right, HurwitzZeta.differentiable_hurwitzZetaEven_sub_hurwitzZetaEven, affine_of_mapsTo_ball_of_norm_dslope_eq_div, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_nhds_right, normSq_ofReal_sub_I_mul_sqrt_one_sub, PeriodPair.derivWeierstrassPExcept_def, continuous_circleMap_inv, sinh_sub_cosh, completedRiemannZeta_one_sub, taylorSeries_eq_on_emetric_ball', GaussianInt.normSq_div_sub_div_lt_one, EulerSine.integral_cos_mul_cos_pow_even, cosh_sub_sinh, LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div, one_div_sub_hasFPowerSeriesOnBall_zero, EulerProduct.exp_tsum_primes_log_eq_tsum, ModularGroup.smul_eq_lcRow0_add, Function.Periodic.exp_decay_sub_of_bounded_at_inf, ModularForm.multipliableLocallyUniformlyOn_eta, tan_two_mul, two_sin, pi_mul_cot_pi_q_exp, PeriodPair.weierstrassPExcept_add, cos_sub_int_mul_two_pi, tan_eq_one_sub_tan_half_sq_div_one_add_tan_half_sq, exp_bound_sq, HurwitzZeta.tendsto_hurwitzZetaEven_sub_one_div_nhds_one, HasDerivAt.cpow_const, hasSum_taylorSeries_on_eball, HurwitzZeta.hurwitzZetaEven_residue_one, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
instZero 📖	CompOp	412 math math: HurwitzZeta.hasSum_int_hurwitzZetaOdd, CStarAlgebra.preimage_inr_Icc_zero_one, TemperedDistribution.lineDerivOpCLM_eq, Real.circleAverage_map_add_const, log_one, MeasureTheory.Lp.toTemperedDistributionCLM_apply, hasSum_mellin_pi_mul_sq', Polynomial.mahlerMeasure_def_of_ne_zero, UnitDisc.coe_eq_zero, restrictScalars_one_smulRight', sin_eq_zero_iff_cos_eq, SchwartzMap.integral_sesq_fourier_fourier, PeriodPair.G_eq_zero_of_odd, Ideal.toCharacterSpace_apply_eq_zero_of_mem, ValueDistribution.proximity_zero, ModularFormClass.hasFPowerSeries_cuspFunction, HurwitzZeta.hasSum_int_cosKernel₀, complexOfReal_hasDerivWithinAt, qExpansion_eq_zero_iff, CuspFormClass.zero_at_infty_comp_ofComplex, Gamma_neg_nat_eq_zero, isCoveringMap_exp, CStarAlgebra.inr_mem_Icc_iff_norm_le, HurwitzZeta.sinZeta_neg_two_mul_nat_sub_one, one_add_cpow_hasFPowerSeriesAt_zero, DirichletCharacter.continuousOn_neg_logDeriv_LFunction_of_nontriv, Unitization.real_cfcₙ_eq_cfc_inr, NumberField.mixedEmbedding.norm_eq_zero_iff', SchwartzMap.integral_bilin_fourierInv_eq, Function.Periodic.cuspFunction_zero_eq_limUnder_nhds_ne, LSeries_injOn, LSeries_eventually_eq_zero_iff', ValueDistribution.proximity_zero_mul_le, cos_eq_zero_iff, UnitDisc.instSMulCommClass_closedBall_circle, TemperedDistribution.lineDerivOp_fourier_eq, UnitDisc.instSMulCommClass_closedBall_left, HasDerivAt.comp_ofReal, circleAverage_log_norm_sub_const₁, HurwitzZeta.cosZeta_neg_two_mul_nat_add_one, LSeries_sub_eventuallyEq_zero_of_LSeries_eventually_eq, TemperedDistribution.smulLeftCLM_compL_smulLeftCLM, sin_int_mul_pi, PeriodPair.weierstrassP_coe, sin_eq_zero_iff, MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.imPart, sum_cauchyPowerSeries_eq_integral, CStarModule.innerSL_apply, UpperHalfPlane.prod_eq_zero_iff, CuspForm.zero_apply, cos_pi_div_two, ModularForm.norm_eq_zero_iff, circleMap_zero_inv, PeriodPair.hasSum_derivWeierstrassPExcept, ValueDistribution.proximity_pow_zero, ZMod.completedLFunction_zero, PeriodPair.weierstrassP_zero, sin_nat_mul_pi, tan_int_mul_pi, CStarMatrix.toCLM_injective, SchwartzMap.integral_bilin_fourierIntegral_eq, tan_nat_mul_pi, OnePoint.isZeroAt_iff_exists_SL2Z, circleMap_sub_center, bernoulliFourierCoeff_zero, VectorFourier.fourierPowSMulRight_eq_comp, tendsto_exp_nhds_zero_iff, TemperedDistribution.smulLeftCLM_neg, Unitization.inr_le_iff, DirichletCharacter.zetaMul_prime_pow_nonneg, SchwartzMap.integral_sesq_fourier_eq, CStarMatrix.norm_def', exp_zero, CStarMatrix.inner_toCLM_conjTranspose_left, TemperedDistribution.fourierInv_lineDerivOp_eq, Orientation.kahler_eq_zero_iff, AddChar.sum_apply_eq_ite, ValueDistribution.proximity_inv, cos_eq_iff_quadratic, LSeries.term_def, mellin_hasDerivAt_of_isBigO_rpow, setIntegral_Ioi_zero_cpow, tan_eq_zero_iff', ModularFormClass.levelOne_neg_weight_eq_zero, norm_circleMap_zero, CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM, TemperedDistribution.instFourierSMul, HurwitzZeta.hurwitzZetaEven_apply_zero, HurwitzZeta.cosZeta_apply_zero, EisensteinSeries.tsum_symmetricIco_linear_sub_linear_add_one_eq_zero, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply_coe, comap_exp_nhds_zero, digamma_zero, NumberField.canonicalEmbedding.integerLattice.inter_ball_finite, HurwitzZeta.hasSum_nat_completedCosZeta, Function.Periodic.boundedAtFilter_cuspFunction, cpow_eq_zero_iff, OnePoint.isZeroAt_infty_iff, EisensteinSeries.tendsto_zero_inv_linear_sub, AddChar.expect_apply_eq_zero_iff_ne_zero, ValueDistribution.characteristic_sub_characteristic_inv_of_ne_zero, ZMod.LFunction_neg_two_mul_nat_sub_one, tan_int_mul_pi_div_two, ArithmeticFunction.one_eq_delta, norm_add_eq_iff, ModularForm.zero_apply, EisensteinSeries.tendsto_e2Summand_atTop_nhds_zero, VonNeumannAlgebra.centralizer_centralizer', ValueDistribution.characteristic_sub_characteristic_inv_at_zero, Quaternion.coeComplex_zero, Mathlib.Meta.Positivity.ofReal_nonneg, AddChar.sum_apply_eq_zero_iff_ne_zero, circleAverage_log_norm_sub_const₀, comap_exp_nhdsNE, circleIntegral.integral_sub_zpow_of_undef, sinh_zero, Polynomial.intervalIntegrable_mahlerMeasure, bernoulli_zero_fourier_coeff, UnitDisc.coe_closedBall_smul, LSeriesSummable_zero, Gammaℝ_eq_zero_iff, CuspFormClass.zero_at_infty, HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one, sin_two_pi, tendsto_self_mul_Gamma_nhds_zero, riemannZeta_neg_two_mul_nat_add_one, SchwartzMap.integral_bilin_fourier_eq, ValueDistribution.proximity_mul_zero_le, arg_lt_pi_div_two_iff, PeriodPair.weierstrassPExcept_zero, SchwartzMap.toTemperedDistributionCLM_apply_apply, map_exp_comap_re_atBot, isEquivalent_sinh, MeasureTheory.SignedMeasure.re_toComplexMeasure, TemperedDistribution.smulLeftCLM_sub, one_add_cpow_hasFPowerSeriesAt_zero, zero_im, OnePoint.isZeroAt_iff_forall_SL2Z, HurwitzZeta.completedHurwitzZetaEven_eq, DirichletCharacter.zetaMul_nonneg, arg_eq_zero_iff_zero_le, Function.locallyFinsuppWithin.toClosedBall_divisor, arg_zero, not_continuousAt_Gamma_zero, sq_nonneg_iff, VonNeumannAlgebra.mem_carrier, has_antideriv_at_fourier_neg, neg_iff, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, one_div_sub_sq_hasFPowerSeriesOnBall_zero, PeriodPair.hasSumLocallyUniformly_weierstrassPExcept, cosh_zero, log_sub_self_isBigO, cfcHom_real_eq_restrict, ValueDistribution.characteristic_sub_characteristic_inv_le, CStarMatrix.toCLM_apply_single, tendsto_exp_comap_re_atBot, VectorFourier.integral_sesq_fourierIntegral_eq_neg_flip, exp_list_sum, one_div_one_sub_pow_hasFPowerSeriesOnBall_zero, hasDerivAt_fourier_neg, nonpos_iff, VectorFourier.integral_fourierIntegral_swap, sqrt_zero, zero_cpow_eq_iff, CStarMatrix.toCLM_apply_eq_sum, tan_zero, ModularFormClass.qExpansion_coeff, riemannZeta_zero, isAddQuotientCoveringMap_exp, inv_zero, Polynomial.logMahlerMeasure_def, normSq_zero, hasMellin_one_Ioc, HadamardThreeLines.interpStrip_eq_of_zero, HurwitzZeta.completedCosZeta_residue_zero, CStarMatrix.toCLM_apply, ValueDistribution.characteristic_mul_zero_eventuallyLE, ValueDistribution.characteristic_pow_zero, UpperHalfPlane.qParam_tendsto_atImInfty, inv_natCast_cpow_ofReal_pos, TemperedDistribution.fourier_lineDerivOp_eq, ModularForm.coe_eq_zero_iff, PositiveLinearMap.gnsStarAlgHom_apply, ValueDistribution.characteristic_sub_characteristic_inv, zero_cpow_eq_nhds, TemperedDistribution.lineDerivOp_toTemperedDistributionCLM_eq, Real.circleAverage_eq_circleAverage_zero_one, Mathlib.Meta.Positivity.ofReal_pos, hasDerivAt_circleMap, UnitDisc.instSMulCommClass_closedBall_right, MeasureTheory.ComplexMeasure.HaveLebesgueDecomposition.rePart, continuousOn_tan, zero_cpow, hasDerivAt_fourier, continuousAt_cpow_zero_of_re_pos, UnitDisc.instIsScalarTower_closedBall, Real.fourier_bilin_convolution_eq, one_div_sub_pow_hasFPowerSeriesOnBall_zero, CStarMatrix.mul_entry_mul_eq_inner_toCLM, circleMap_zero, zero_lt_real, SchwartzMap.integral_sesq_fourierIntegral_eq, deriv_circleMap, Polynomial.X_pow_sub_X_sub_one_irreducible_aux, LSeries.term_delta, HurwitzZeta.hurwitzOddFEPair_f₀, circleMap_zero_mul, isCoveringMapOn_exp, riemannZeta_pos_of_one_lt, ValueDistribution.characteristic_mul_zero_le, ofReal_eq_zero, ofReal_zero, TemperedDistribution.smulLeftCLM_add, PeriodPair.derivWeierstrassP_coe, locallyFinsuppWithin.logCounting_divisor, Polynomial.sum_sq_norm_coeff_eq_circleAverage, SlashInvariantForm.coe_zero, exp_sub_sum_range_succ_isLittleO_pow, log_div_self, HasDerivAt.ofReal_comp, PeriodPair.sumInvPow_zero, cos_zero, circleMap_zero_div, EisensteinSeries.tendsto_zero_inv_linear, Unitization.nnreal_cfcₙ_eq_cfc_inr, circleAverage_log_norm_add_const_eq_posLog, UnitDisc.coe_smul_closedBall, jacobiTheta₂_undef, isConformalMap_complex_linear_conj, exp_sub_sum_range_isBigO_pow, isOpenQuotientMap_zpow_compl_zero, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, Function.Periodic.invQParam_tendsto, cfcₙHom_real_eq_restrict, UnitDisc.mk_zero, MeasureTheory.ComplexMeasure.im_apply, Function.locallyFinsuppWithin.logCounting_divisor_eq_circleAverage_sub_const, UnitDisc.mk_eq_zero, EisensteinSeries.tsum_tsum_symmetricIco_sub_eq, PeriodPair.hasSumLocallyUniformly_derivWeierstrassPExcept, Real.hasDerivAt_fourierChar, span_fourierLp_closure_eq_top, ZMod.LFunction_neg_two_mul_nat_add_one, circleIntegral.integral_sub_zpow_of_ne, LindemannWeierstrass.integral_exp_mul_eval, inner_map_self_eq_zero, GaussianFourier.tendsto_verticalIntegral, EisensteinSeries.D2_one, norm_sub_one_sq_eqOn_sphere, logTaylor_zero, abs_arg_lt_pi_div_two_iff, ValueDistribution.proximity_zero_of_complexValued, not_le_zero_iff, eq_zero_cpow_iff, pos_iff, not_differentiableAt_Gamma_zero, DirichletCharacter.Even.LFunction_neg_two_mul_nat, ValueDistribution.characteristic_zero_mul_eventually_le, Function.Periodic.qParam_tendsto, Gammaℝ_residue_zero, EulerSine.antideriv_cos_comp_const_mul, re_eq_neg_norm, NumberField.mixedEmbedding.forall_normAtPlace_eq_zero_iff, hasFPowerSeriesOnBall_cuspFunction, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat_sub_one, ModularFormClass.qExpansion_coeff_eq_circleIntegral, CuspForm.coe_zero, one_div_one_sub_cpow_hasFPowerSeriesOnBall_zero, Function.Periodic.eventually_differentiableAt_cuspFunction_nhds_ne_zero, circleAverage_log_norm_sub_const_eq_posLog, EulerSine.antideriv_sin_comp_const_mul, zero_le_real, MeasureTheory.charFun_zero_measure, TemperedDistribution.instFourierInvSMul, HurwitzZeta.hurwitzOddFEPair_g₀, sin_pi, hasDerivAt_ofReal_cpow_const, NumberField.mixedEmbedding.unit_smul_eq_zero, one_add_cpow_hasFPowerSeriesOnBall_zero, norm_sub_eq_iff, tan_eq_zero_iff, UnitDisc.coe_zero, LSeries_zero, VectorFourier.integral_bilin_fourierIntegral_eq_flip, TemperedDistribution.instLineDerivSMulComplex, ValueDistribution.characteristic_zero_mul_le, circleAverage_log_norm_sub_const₂, sq_nonpos_iff, one_div_one_sub_sq_hasFPowerSeriesOnBall_zero, UpperHalfPlane.mul_eq_zero_iff, ZMod.LFunction_apply_zero_of_even, IsConservativeOn.hasDerivAt_wedgeIntegral, HurwitzZeta.completedCosZeta_eq, VonNeumannAlgebra.coe_toStarSubalgebra, jacobiTheta_eq_jacobiTheta₂, zero_re, sameRay_iff, PeriodPair.hasSum_weierstrassPExcept, compl_Iic_zero, jacobiTheta₂'_undef, not_lt_zero_iff, GaussianInt.toComplex_zero, log_sub_logTaylor_isBigO, Function.Periodic.tendsto_at_I_inf, TemperedDistribution.derivCLM_toTemperedDistributionCLM_eq, cos_eq_zero_iff_sin_eq, ModularFormClass.cuspFunction_apply_zero, UnitDisc.instSMulCommClass_circle_closedBall, cpow_def, LSeries_eq_zero_of_abscissaOfAbsConv_eq_top, CuspFormClass.cuspFunction_apply_zero, hasDerivAt_ofReal_cpow_const', continuous_tan, MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff, MeasureTheory.Lp.ker_toTemperedDistributionCLM_eq_bot, MeasureTheory.Lp.toTemperedDistribution_toLp_eq, tanh_zero, ValueDistribution.logCounting_zero_sub_logCounting_top_eq_circleAverage_sub_const, PeriodPair.derivWeierstrassPExcept_zero_zero, CuspFormClass.zero_at_infty_slash, bernoulliFourierCoeff_recurrence, DirichletCharacter.Even.LFunction_neg_two_mul_nat_add_one, TemperedDistribution.smulLeftCLM_smulLeftCLM_apply, RCLike.to_complex_nonneg_iff, isOpenQuotientMap_pow_compl_zero, ModularFormClass.analyticAt_cuspFunction_zero, MeasureTheory.ComplexMeasure.re_apply, logTaylor_at_zero, ProbabilityTheory.complexMGF_undef, TemperedDistribution.lineDerivOp_fourierInv_eq, circleMap_zero_pow, complexOfReal_hasDerivAt, ModularFormClass.differentiableOn_cuspFunction_ball, CStarAlgebra.inr_mem_Icc_iff_nnnorm_le, sin_zero, TemperedDistribution.smulLeftCLM_const, range_exp, DirichletCharacter.Odd.LFunction_neg_two_mul_nat_sub_one, Real.circleAverage_zero_one_congr_inv, TemperedDistribution.smulLeftCLM_sum, nonneg_iff, one_add_cpow_hasFPowerSeriesOnBall_zero, IsExpCmpFilter.eventually_ne, MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure, hasMellin_cpow_Ioc, one_div_one_sub_hasFPowerSeriesOnBall_zero, image_exp_Ioc_eq_sphere, HurwitzZeta.completedHurwitzZetaEven_residue_zero, TemperedDistribution.smulLeftCLM_smul, deriv_circleMap_eq_zero_iff, qExpansion_zero, isConformalMap_complex_linear, LSeries.eq_zero_of_not_LSeriesSummable, TemperedDistribution.derivCLM_apply_apply, ValueDistribution.proximity_coe, Function.Periodic.differentiableAt_cuspFunction_zero, eventually_eq_or_eq_zero_of_isLocalMin_norm, WeakDual.CharacterSpace.exists_apply_eq_zero, DirichletCharacter.LSeriesSummable_zetaMul, ModularForm.coe_zero, AddChar.expect_apply_eq_ite, re_eq_norm, torusMap_sub_center, VectorFourier.hasFDerivAt_fourierChar_smul, one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnBall_zero, NumberField.mixedEmbedding.logMap_zero, cpow_zero, log_zero, OnePoint.isZeroAt_iff, UnitAddTorus.span_mFourierLp_closure_eq_top, zero_notMem_slitPlane, Gamma_eq_zero_iff, range_exp_mul_I, Polynomial.fourierCoeff_toAddCircle, hasFPowerSeriesOnBall_ofScalars_mul_add_zero, WeakFEPair.Λ_residue_zero, Gamma_zero, SlashInvariantForm.wt_eq_zero_of_eq_const, ValueDistribution.proximity_top, neg_re_eq_norm, PositiveLinearMap.gnsNonUnitalStarAlgHom_apply, re_nonneg_iff_nonneg, LindemannWeierstrass.hasDerivAt_cexp_mul_sumIDeriv, EisensteinSeries.D2_T, UnitDisc.instIsScalarTower_closedBall_closedBall, normSq_eq_zero, mem_slitPlane_iff_not_le_zero, TemperedDistribution.laplacianCLM_apply, ValueDistribution.proximity_sub_proximity_inv_eq_circleAverage, DirichletCharacter.continuousOn_neg_logDeriv_LFunctionTrivChar₁, MeasureTheory.Lp.toTemperedDistribution_smul_eq, cpow_ne_zero_iff, one_div_sub_hasFPowerSeriesOnBall_zero, Function.Periodic.exp_decay_sub_of_bounded_at_inf, Unitization.inr_nonneg_iff, CStarMatrix.norm_def, PeriodPair.derivWeierstrassP_zero, isEquivalent_sin, TemperedDistribution.smulLeftCLM_apply_apply, arg_eq_pi_iff_lt_zero, LSeries.term_zero, MeasureTheory.SignedMeasure.im_toComplexMeasure, CStarMatrix.inner_toCLM_conjTranspose_right, GaussianInt.toComplex_eq_zero, CStarMatrix.toCLM_apply_single_apply, tendsto_exp_comap_re_atBot_nhdsNE, restrictScalars_toSpanSingleton', circleMap_zero_zpow, Polynomial.fourierCoeff_toAddCircle_eq_zero_of_lt_zero, NumberField.mixedEmbedding.convexBodySumFun_eq_zero_iff, fourierCoeff_fourier
normSq 📖	CompOp	58 math math: UnitDisc.normSq_lt_one, normSq_pos, normSq_natCast, UpperHalfPlane.normSq_pos, norm_mul_self_eq_normSq, UpperHalfPlane.im_smul_eq_div_normSq, UpperHalfPlane.moebius_im, ModularGroup.one_lt_normSq_T_zpow_smul, normSq_ratCast, normSq_mk, GaussianInt.normSq_le_normSq_of_re_le_of_im_le, normSq_mul, sq_norm, normSq_I, tendsto_normSq_cocompact_atTop, normSq_add, re_sq_le_normSq, div_im, normSq_eq_norm_sq, normSq_sub, inv_re, UpperHalfPlane.normSq_denom_pos, inv_def, normSq_ofNat, Algebra.norm_complex_eq, normSq_ofReal_add_I_mul_sqrt_one_sub, normSq_zero, normSq_conj, continuous_normSq, ModularGroup.tendsto_normSq_coprime_pair, normSq_intCast, div_re, Orientation.normSq_kahler, normSq_inv, normSq_div, normSq_ofReal, Algebra.norm_complex_apply, norm_def, GaussianInt.intCast_real_norm, normSq_nonneg, ModularGroup.im_smul_eq_div_normSq, normSq_add_mul_I, mul_conj, inv_im, Circle.normSq_coe, normSq_eq_conj_mul_self, normSq_apply, RCLike.normSq_to_complex, im_sq_le_normSq, normSq_neg, UpperHalfPlane.c_mul_im_sq_le_normSq_denom, range_normSq, UpperHalfPlane.smulAux'_im, normSq_ofReal_sub_I_mul_sqrt_one_sub, normSq_eq_zero, GaussianInt.normSq_div_sub_div_lt_one, GaussianInt.intCast_complex_norm, normSq_one
ofReal 📖	CompOp	847 math math: ArithmeticFunction.vonMangoldt.residueClass_eq, HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip_eps, HurwitzZeta.hasSum_int_hurwitzZetaOdd, cos_sub_pi, Real.hasDerivAt_fourier, verticalSegment_eq, ProbabilityTheory.integrable_cexp_mul_of_re_mem_integrableExpSet, EulerSine.integral_cos_mul_cos_pow, ofNat_log, tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero, log_I, re_mul_ofReal, div_ofReal_re, ProbabilityTheory.isGaussian_iff_gaussian_charFun, nnnorm_exp_I_mul_ofReal, log_neg_one, Measurable.complex_ofReal, completedZeta_eq_tsum_of_one_lt_re, coe_realPart, GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I, Real.hasFDerivAt_fourierChar_neg_bilinear_left, EisensteinSeries.hasSum_e2Summand_symmetricIcc, mk_eq_add_mul_I, Gammaℝ_def, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, ofReal_arctan, sin_pi_div_two_sub, HurwitzZeta.hasSum_int_cosKernel₀, exp_eq_exp_iff_exists_int, HurwitzZeta.hasSum_int_hurwitzZetaEven, re_eq_add_conj, subset_slitPlane_iff_of_subset_sphere, HurwitzZeta.LSeriesHasSum_cos, ofReal_eq_one, Real.fourierIntegralInv_eq', tan_nat_mul_pi_sub, integral_boundary_rect_eq_zero_of_differentiableOn, MeasureTheory.charFunDual_apply, Polynomial.Chebyshev.complex_ofReal_eval_S, ofRealLI_apply, GammaIntegral_eq_mellin, isLittleO_exp_neg_mul_sq_cocompact, ProbabilityTheory.hasGaussianLaw_iff_charFunDual_map_eq, HurwitzZeta.cosZeta_two_mul_nat', real_smul, summableLocallyUniformlyOn_iteratedDerivWithin_cexp, realPart_ofReal, cos_eq_one_iff, ModularForm.constℝ_apply, equivRealProdAddHom_symm_apply, ModularForm.eta_q_eq_pow, ZMod.LFunction_one_sub, tsum_exp_neg_mul_int_sq, not_integrableOn_Ioi_cpow, Real.fderiv_fourierChar_neg_bilinear_right_apply, SchwartzMap.lineDerivOp_fourier_eq, arg_cos_add_sin_mul_I_coe_angle, tan_pi_sub, circleMap_neg_pi_div_two, GaussianFourier.integral_cexp_neg_mul_sq_norm, LSeries_eventually_eq_zero_iff', angle_exp_one, cos_ofReal_re, circleIntegral.integral_sub_inv_of_mem_ball, hasSum_one_div_nat_pow_mul_fourier, re_sqrt_ofReal, betaIntegral_scaled, cos_eq_zero_iff, GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace, cos_antiperiodic, ofReal_cosh_ofReal_re, cos_add_nat_mul_two_pi, TemperedDistribution.lineDerivOp_fourier_eq, WeakFEPair.symm_Λ₀_eq, HasProdUniformlyOn_sineTerm_prod_on_compact, HurwitzZeta.cosZeta_two_mul_nat, norm_eq_one_iff', sin_eq_neg_one_iff, UpperHalfPlane.specialLinearGroup_apply, SlashInvariantForm.coe_constℝ, cpow_mul_ofReal_nonneg, riemannZeta_two, ProbabilityTheory.integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux, fourierIntegral_gaussian_innerProductSpace', sqrt_I, fourier_gaussian_pi', iteratedDerivWithin_tsum_cexp_eq, jacobiTheta₂_functional_equation, intervalIntegral.integral_ofReal, ProbabilityTheory.hasGaussianLaw_iff_charFun_map_eq, sin_int_mul_pi, GaussianFourier.integral_cexp_neg_mul_sq_norm_add, norm_cexp_neg_mul_sq, ModularForm.coe_constℝ, Polynomial.Chebyshev.complex_ofReal_eval_T, sin_eq_zero_iff, ofReal_sin_ofReal_re, sum_cauchyPowerSeries_eq_integral, tan_pi_div_two_sub, DirichletCharacter.LSeries_twist_vonMangoldt_eq, Real.norm_exp_I_mul_ofReal_sub_one_le, Real.vector_fourierIntegral_eq_integral_exp_smul, HurwitzZeta.evenKernel_def, cfc_complex_eq_real, Real.fourier_real_eq_integral_exp_smul, norm_one_add_mul_inv_le, HurwitzZeta.jacobiTheta₂'_functional_equation', norm_cos_add_sin_mul_I, cos_pi_div_two, hasSum_nat_jacobiTheta, Real.fourier_deriv, DirichletCharacter.norm_LSeries_product_ge_one, digamma_one_half, ZMod.stdAddChar_coe, DifferentiableOn.circleIntegral_sub_inv_smul, HarmonicAt.analyticAt_complex_partial, ModularGroup.coe_apply_complex, Real.nnnorm_exp_I_mul_ofReal_sub_one_le, WeakFEPair.h_feq, integrableOn_Ioi_norm_cpow_of_lt, sin_nat_mul_pi, tan_int_mul_pi, ModularForm.tsum_logDeriv_eta_q, SlashInvariantForm.constℝ_toFun, integral_circleTransform, ofReal_tanh, HurwitzZeta.oddKernel_def', fourier_gaussian_innerProductSpace, tan_nat_mul_pi, isPrimitiveRoot_exp_rat, VectorFourier.fourierPowSMulRight_eq_comp, Polynomial.cyclotomic.eval_apply_ofReal, mem_rootsOfUnity, MeasureTheory.charFun_apply, sin_int_mul_two_pi_sub, cosh_ofReal_im, cos_nat_mul_two_pi_sub_pi, hasDerivAt_GammaIntegral, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable, deriv_ofReal_cpow_const, starConvex_ofReal_slitPlane, ArithmeticFunction.LSeries_positive, HurwitzZeta.hasSum_nat_sinZeta, exp_antiperiodic, MeasureTheory.charFun_apply_real, StrongFEPair.functional_equation, fourierCoeff_bernoulli_eq, exp_add_pi_mul_I, log_exp_eq_re_add_toIocMod, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace, circleMap_eq_circleMap_iff, EulerSine.integral_cos_mul_cos_pow_aux, SchwartzMap.fourier_lineDerivOp_eq, equivRealProd_symm_apply, Polynomial.Chebyshev.complex_ofReal_eval_C, UpperHalfPlane.re_add_im, Filter.tendsto_ofReal_iff, TemperedDistribution.fourierInv_lineDerivOp_eq, mul_conj', RCLike.complexRingEquiv_apply, ofReal_cot, arg_cos_add_sin_mul_I, riemannZeta_one_sub, integrableOn_Ioi_cpow_iff, integral_exp_mul_complex_Ioi, setIntegral_Ioi_zero_cpow, ProbabilityTheory.integrable_rpow_abs_mul_cexp_of_re_mem_interior_integrableExpSet, deriv_norm_ofReal_cpow, tan_eq_zero_iff', cos_nat_mul_two_pi, ofReal_zpow, NumberField.tendsto_sub_one_mul_dedekindZeta_nhdsGT, ofReal_cpow, cauchyPowerSeries_apply, IsSelfAdjoint.coe_mem_spectrum_complex, ofReal_mul', ofReal_log, riemannZeta_im_eq_zero_of_one_lt, HurwitzZeta.hasSum_hurwitzZeta_of_one_lt_re, tan_ofReal_re, sin_sub_nat_mul_two_pi, DiffContOnCl.circleIntegral_sub_inv_smul, ofReal_cpow_of_nonpos, sin_nat_mul_two_pi_sub, exp_ofReal_re, ofReal_cos_ofReal_re, integrableOn_Ioi_norm_cpow_iff, HurwitzZeta.expZeta_one_sub, arg_ofReal_of_nonneg, ArithmeticFunction.vonMangoldt.residueClass_apply, LSeriesSummable_of_bounded_of_one_lt_real, fourierIntegral_gaussian_pi, ProbabilityTheory.complexMGF_mul_I, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow, ArithmeticFunction.LSeries_positive_of_differentiable_of_eqOn, Real.circleAverage_eq_circleIntegral, norm_exp_I_mul_ofReal_sub_one, integral_exp_mul_complex, ArithmeticFunction.vonMangoldt.abscissaOfAbsConv_residueClass_le_one, ofReal_comp_sub, arg_mul_cos_add_sin_mul_I_eq_toIocMod, arg_cos_add_sin_mul_I_sub, equivRealProdCLM_symm_apply, GammaIntegral_ofReal, mellinInv_eq_fourierIntegralInv, cot_series_rep', natCast_log, cos_pi_sub, norm_ofReal_cpow_eventually_eq_atTop, exists_norm_eq_mul_self, Real.Angle.coe_toCircle, hasDerivAt_Gammaℝ_one, HadamardThreeLines.mem_verticalClosedStrip_of_scale_id_mem_verticalClosedStrip, ofReal_neg, circleMap_pi_div_two, riemannZeta_four, HurwitzZeta.hasSum_nat_completedCosZeta, cos_sub_two_pi, partialGamma_add_one, sin_sub_two_pi, sin_sub_pi_div_two, mellinInv_eq_fourierInv, norm_eq_one_iff, exp_ofReal_mul_I_re, fourier_gaussian_innerProductSpace', HurwitzZeta.hasSum_nat_hurwitzZetaEven, arg_mul_cos_add_sin_mul_I, HurwitzZeta.hasSum_int_cosKernel, IsExpCmpFilter.isLittleO_exp_cpow, integral_cpow, IsSelfAdjoint.val_re_map_spectrum, Real.tendsto_integral_gaussian_smul', HurwitzZeta.hurwitzZeta_neg_nat, exp_neg_pi_div_two_mul_I, GaussianFourier.integrable_cexp_neg_mul_sum_add, AnalyticOnNhd.eq_const_add_im_mul_I_of_re_eq_const, tendsto_riemannZeta_sub_one_div, ZetaAsymptotics.tendsto_Gamma_term_aux, ofReal_balance, EisensteinSeries.D2_S, Polynomial.int_cyclotomic_rw, tan_int_mul_pi_div_two, cpow_ofReal_re, LSeries_eq_mul_integral_of_nonneg, EisensteinSeries.G2_S_transform, HurwitzZeta.oddKernel_def, isIntegral_exp_rat_mul_pi_mul_I, DiffContOnCl.deriv_eq_smul_circleIntegral, DifferentiableOn.deriv_eq_smul_circleIntegral, log_ofReal_mul, tan_ofReal_im, cos_int_mul_two_pi_sub, HarmonicAt.differentiableAt_complex_partial, tendsto_logDeriv_euler_sin_div, Real.fourierIntegral_iteratedDeriv, ProbabilityTheory.IsGaussian.charFun_eq', logDeriv_sin_div_eq_cot, HurwitzZeta.cosZeta_one_sub, EulerSine.sin_pi_mul_eq, integral_mul_cexp_neg_mul_sq, ofReal_sinh_ofReal_re, isPrimitiveRoot_iff, isometry_ofReal, norm_cpow_real, exp_periodic, Mathlib.Meta.Positivity.ofReal_nonneg, isTheta_ofReal, norm_exp_mul_sq_le, WeakFEPair.Λ₀_eq, exp_nat_mul_two_pi_mul_I, log_exp_eq_sub_toIocDiv, ProbabilityTheory.complexMGF_id_mul_I, deriv_Gamma_nat, isAlgebraic_sin_rat_mul_pi, cfcₙ_real_eq_complex, ofReal_intCast, ofReal_add, ProbabilityTheory.complexMGF_gaussianReal, isPrimitiveRoot_exp_of_isCoprime, DiffContOnCl.circleIntegral_one_div_sub_center_pow_smul, nhdsWithin_lt_le_nhdsWithin_stolzSet, hasSum_ofReal, ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div, HadamardThreeLines.interpStrip_scale, intervalIntegral.intervalIntegrable_cpow, integral_exp_mul_I_eq_sinc, ofReal_cot_ofReal_re, sin_two_pi, UpperHalfPlane.norm_exp_two_pi_I_lt_one, HurwitzZeta.hasSum_int_cosZeta, LSeries.term_nonneg, arg_exp_mul_I, WeakFEPair.h_feq', cos_add_pi, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_Gammaℝ, tendsto_norm_tan_atTop, ProbabilityTheory.IsGaussian.charFunDual_eq_of_forall_strongDual_eq_zero, BoundedContinuousFunction.innerProbChar_apply, ofReal_prod, hasDerivAt_Gammaℂ_one, sub_conj, sin_add_two_pi, IsSelfAdjoint.cfc_arg, EisensteinSeries.hasSum_e2Summand_symmetricIco, ModularForm.eta_q_eq_cexp, ArithmeticFunction.convolution_vonMangoldt_const_one, intervalIntegral.integrableOn_Ioo_cpow_iff, arg_mul_cos_add_sin_mul_I_coe_angle, circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const, EisensteinSeries.q_expansion_riemannZeta, SlashInvariantFormClass.periodic_comp_ofComplex, integrableOn_Ioi_deriv_ofReal_cpow, HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat, Real.fourierIntegral_real_eq_integral_exp_smul, fourier_coe_apply, Real.deriv_fourierIntegral, sqrt_of_nonneg, integral_exp_mul_I_eq_sin, integral_boundary_rect_of_hasFDerivAt_real_off_countable, ArithmeticFunction.convolution_vonMangoldt_zeta, HurwitzZeta.hasSum_nat_cosZeta, hasSum_cauchyPowerSeries_integral, tan_sub_int_mul_pi, sin_eq_sin_iff, ProbabilityTheory.isGaussian_iff_charFun_eq, ofReal_one, HasProdLocallyUniformlyOn_euler_sin_prod, NormedSpace.ofReal_exp_ℝ_ℝ, sin_pi_sub, ProbabilityTheory.beta_eq_betaIntegralReal, has_antideriv_at_fourier_neg, ofReal_nnratCast, HadamardThreeLines.interpStrip_eq_of_pos, HurwitzZeta.hasSum_expZeta_of_one_lt_re, HurwitzZeta.hasSum_int_completedHurwitzZetaOdd, inv_def, ofRealCLM_apply, two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable, cos_sub_nat_mul_two_pi, cos_int_mul_two_pi_add_pi, mul_cpow_ofReal_nonneg, summable_ofReal, hasDerivAt_fourier_neg, ofReal_exp, SchwartzMap.fourierInv_lineDerivOp_eq, tendsto_euler_sin_prod, MeasureTheory.integral_charFun_Icc, normSq_ofReal_add_I_mul_sqrt_one_sub, exp_ofReal_mul_I_im, EulerSine.integral_sin_mul_sin_mul_cos_pow_eq, eq_coe_norm_of_nonneg, ProbabilityTheory.IsGaussian.charFunDual_eq, tsum_exp_neg_quadratic, ofReal_comp_add, norm_real, conj_mul', isAddQuotientCoveringMap_exp, SpectrumRestricts.real_iff, cot_pi_eq_exp_ratio, EisensteinSeries.auxbound1, conj_eq_iff_re, exp_ofReal_im, isIntegral_two_mul_sin_rat_mul_pi, ofReal_sum, HurwitzZeta.hurwitzZetaEven_one_sub, HurwitzZeta.hasSum_int_sinKernel, ofReal_comp_mul, DirichletCharacter.LSeriesSummable_twist_vonMangoldt, QuasispectrumRestricts.real_iff, cot_series_rep, norm_of_nonneg, sin_antiperiodic, HurwitzZeta.LSeriesHasSum_exp, DirichletCharacter.convolution_twist_vonMangoldt, ofReal_comp_balance, bernoulliFourierCoeff_eq, ofReal_nsmul, norm_mul_cos_add_sin_mul_I, isPrimitiveRoot_exp_of_coprime, tan_sub_nat_mul_pi, inv_natCast_cpow_ofReal_pos, ModularForm.logDeriv_eta_eq_E2, TemperedDistribution.fourier_lineDerivOp_eq, sin_eq_one_iff, equivRealProdLm_symm_apply, MeasureTheory.charFunDual_map_const_add, MeasureTheory.charFunDual_map_add_const, add_conj, HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat, iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_zpow, Real.fourier_iteratedDeriv, cos_periodic, WeakFEPair.hf_zero, exp_pi_div_two_mul_I, continuousAt_ofReal_cpow, LinearMap.isPositive_iff_complex, VectorFourier.fourierSMulRight_apply, integral_cexp_quadratic, Real.circleAverage_eq_circleAverage_zero_one, circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, orthonormalBasisOneI_repr_symm_apply, Mathlib.Meta.Positivity.ofReal_pos, tanh_ofReal_re, HurwitzZeta.sinZeta_two_mul_nat_add_one', tan_sub_pi, sinh_ofReal_re, Gammaℂ_def, norm_mul_exp_arg_mul_I, tan_add_pi, NumberField.mixedEmbedding.stdBasis_repr_eq_matrixToStdBasis_mul, ofReal_expect, isPrimitiveRoot_exp_rat_of_even_num, hasDerivAt_fourier, Filter.Tendsto.ofReal, sin_ofReal_re, isPrimitiveRoot_exp, ProbabilityTheory.isGaussian_iff_gaussian_charFunDual, isIntegral_exp_neg_rat_mul_pi_mul_I, arg_eq_arg_iff, HurwitzZeta.hasSum_nat_completedSinZeta, circleMap_zero, integrableOn_exp_mul_complex_Iic, zero_lt_real, tanh_ofReal_im, LSeries_eq_mul_integral, Real.fourierIntegral_eq', ofReal_tan_ofReal_re, hasFPowerSeriesOn_cauchy_integral, NumberField.InfinitePlace.embedding_of_isReal_apply, circleIntegral_div_sub_of_differentiable_on_off_countable, contDiffOn_tsum_cexp, Gammaℂ_add_one, cos_add_pi_div_two, re_eq_ofReal_of_isSelfAdjoint, EisensteinSeries.auxbound2, Orientation.kahler_apply_self, fourierIntegral_gaussian_innerProductSpace, ProbabilityTheory.HasGaussianLaw.charFun_map_eq, cexp_neg_quadratic_isLittleO_abs_rpow_cocompact, ArithmeticFunction.LSeries_vonMangoldt_eq, Quaternion.coeComplex_coe, EisensteinSeries.qExpansion_identity_pnat, Real.iteratedDeriv_fourier, riemannZeta_pos_of_one_lt, UpperHalfPlane.σ_ofReal, ofReal_eq_zero, ofReal_zero, cot_pi_mul_contDiffWithinAt, norm_exp_ofReal, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', div_ofReal, ofReal_exp_ofReal_re, tendsto_one_add_div_cpow_exp, LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_and_nonneg, jacobiTheta₂'_add_left', jacobiTheta₂'_functional_equation, isTheta_ofReal_right, riemannZeta_one, ProbabilityTheory.integrable_cexp_mul_of_re_mem_interior_integrableExpSet, HasDerivAt.ofReal_comp, normSq_ofReal, DifferentiableAt.ofReal_cpow_const, isBigO_ofReal_left, sin_pi_div_two, sin_periodic, differentiable_on_off_countable_deriv_eq_smul_circleIntegral, rotationOf_coe, cos_sub_pi_div_two, selfAdjoint.mem_spectrum_eq_re, NumberField.ComplexEmbedding.IsReal.coe_embedding_apply, norm_add_mul_I, tan_int_mul_pi_sub, tendsto_logDeriv_euler_cot_sub, mellin_comp_rpow, DirichletCharacter.norm_LFunction_product_ge_one, IsExpCmpFilter.isLittleO_zpow_mul_exp, measurableEquivPi_symm_apply, HurwitzZeta.hasSum_nat_hurwitzZetaOdd, betaIntegral_convergent, norm_of_nonneg', coe_algebraMap, enorm_exp_I_mul_ofReal, cos_int_mul_two_pi_sub_pi, cos_add_two_pi, ofReal_comp_pow, Real.hasDerivAt_fourierChar, isUniformEmbedding_ofReal, LSeries.iteratedDeriv_alternating, exists_norm_mul_eq_self, MellinConvergent.comp_rpow, WeakFEPair.functional_equation₀, ofReal_injective, Real.tendsto_integral_cexp_sq_smul, Real.enorm_exp_I_mul_ofReal_sub_one_le, EisensteinSeries.q_expansion_bernoulli, UpperHalfPlane.coe_specialLinearGroup_apply, HurwitzZeta.cosKernel_def, ofReal_mem_slitPlane, wedgeIntegral_add_wedgeIntegral_eq, AEMeasurable.complex_ofReal, sqrt_neg_I, HurwitzZeta.sinZeta_two_mul_nat_add_one, integral_boundary_rect_eq_zero_of_differentiable_on_off_countable, sqrt_eq_real_add_ite, EulerSine.tendsto_integral_cos_pow_mul_div, coe_selfAdjointEquiv, LSeries.term_pos, completedRiemannZeta₀_one, fourierCoeffOn_of_hasDerivAt_Ioo, imaginaryPart_ofReal, circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto, SchwartzMap.lineDerivOp_fourierInv_eq, arg_mul_real, sin_eq, Real.fourierChar_apply, HadamardThreeLines.interpStrip_eq_of_mem_verticalStrip, coe_smul, integrableOn_Ioi_cpow_of_lt, GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I, EulerSine.antideriv_cos_comp_const_mul, sinh_ofReal_im, mellin_convergent_iff_norm, normSq_add_mul_I, Orientation.kahler_mul, GaussianFourier.integral_cexp_neg_mul_sum_add, continuous_ofReal_cpow_const, ModularFormClass.qExpansion_coeff_eq_circleIntegral, ProbabilityTheory.isGaussian_iff_charFunDual_eq, fourierIntegral_gaussian, LSeries.tendsto_cpow_mul_atTop, ArithmeticFunction.vonMangoldt.LSeries_residueClass_eq, im_mul_ofReal, integrable_pow_mul_norm_one_add_mul_inv, sin_add_int_mul_two_pi, LSeriesSummable_of_sum_norm_bigO_and_nonneg, HurwitzZeta.hasSum_nat_hurwitzZetaEven_of_mem_Icc, im_eq_sub_conj, EulerSine.antideriv_sin_comp_const_mul, zero_le_real, ofReal_natCast, HurwitzZeta.hurwitzZeta_one_sub, WeakFEPair.hf_modif_FE, integral_gaussian_sq_complex, ofReal_qsmul, ContinuousLinearMap.isPositive_iff_complex, DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul, sin_pi, continuous_ofReal, ofReal_zsmul, exp_pi_mul_I, isBigO_deriv_ofReal_cpow_const_atTop, isPrimitiveRoot_exp_rat_of_odd_num, tan_add_int_mul_pi, hasDerivAt_ofReal_cpow_const, ofReal_sinh, isIntegral_two_mul_cos_rat_mul_pi, LSeries.positive, selfAdjointEquiv_symm_apply, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq_fun, ofRealHom_eq_coe, Gammaℝ_one_sub_mul_Gammaℝ_one_add, tan_add_nat_mul_pi, HurwitzZeta.hurwitzOddFEPair_f, LSeries.tendsto_atTop, exp_sub_pi_mul_I, SchwartzMap.fderivCLM_fourier_eq, Polynomial.Chebyshev.complex_ofReal_eval_U, tan_eq_zero_iff, Gammaℝ_add_two, GammaSeq_eq_approx_Gamma_integral, nnnorm_exp_ofReal_mul_I, riemannZeta_re_pos_of_one_lt, AlgHom.map_coe_real_complex, cexp_neg_quadratic_isLittleO_rpow_atTop, NormedSpace.of_real_exp_ℝ_ℝ, cpow_ofReal_im, integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real, ProbabilityTheory.integrable_pow_abs_mul_cexp_of_re_mem_interior_integrableExpSet, euler_sineTerm_tprod, ofReal_cos, ProbabilityTheory.integrable_pow_mul_cexp_of_re_mem_interior_integrableExpSet, Function.Complex.hasTemperateGrowth_ofReal, ofReal_comp_nsmul, WeakFEPair.functional_equation, GaussianFourier.integrable_cexp_neg_mul_sq_norm_add, cos_eq, ofReal_div, IsSelfAdjoint.mem_spectrum_eq_re, ofReal_comp_zsmul, ProbabilityTheory.complexMGF_id_gaussianReal, mul_conj, im_ofReal_mul, ofReal_inj, exp_rat_mul_pi_mul_I_pow_two_mul_den, digamma_one, VectorFourier.fourierPowSMulRight_apply, LSeriesSummable_lt_re_of_abscissaOfAbsConv_lt_re, MeasureTheory.charFunDual_dirac, ofReal_re, EisensteinSeries.G2_eq_tsum_cexp, hasDerivAt_Gamma_one, HadamardThreeLines.eventuallyle, circleIntegral_one_div_sub_center_pow_smul_of_differentiable_on_off_countable, sin_sub_pi, EisensteinSeries.qExpansion_identity, GaussianFourier.integral_cexp_neg_sum_mul_add, selfAdjoint.val_re_map_spectrum, RCLike.ofReal_eq_complex_ofReal, cos_pi_div_two_sub, tsum_eisSummand_eq_tsum_sigma_mul_cexp_pow, arg_real_mul, ofReal_sub, rootsOfUnityCircleEquiv_comp_rootsOfUnityAddChar_val, HurwitzZeta.hasSum_int_sinZeta, UpperHalfPlane.coe_vadd, conj_eq_iff_real, ZMod.toCircle_intCast, iteratedDerivWithin_cot_sub_inv_eq_add_mul_tsum, ProbabilityTheory.complexMGF_ofReal, ZMod.toCircle_natCast, Real.deriv_fourierChar, exp_eq_exp_re_mul_sin_add_cos, jacobiTheta_eq_tsum_nat, canLift, RCLike.norm_to_complex, isBigO_ofReal_right, Function.Periodic.qParam_left_inv_mod_period, ofReal_sin, Orientation.kahler_apply_apply, div_ofReal_im, hasDerivAt_ofReal_cpow_const', hasDerivAt_Gamma_one_half, normSq_eq_conj_mul_self, ModularForm.slash_def, Real.tendsto_integral_gaussian_smul, sin_sub_int_mul_two_pi, bernoulliFourierCoeff_recurrence, Gamma_mul_Gamma_add_half, ArithmeticFunction.vonMangoldt.LFunctionResidueClassAux_real, EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq, sin_two_pi_sub, integral_complex_ofReal, NumberField.mixedEmbedding.norm_real, VectorFourier.fourierIntegral_probChar, sin_ofReal_im, RCLike.to_complex_nonneg_iff, integral_Ioi_cpow_of_lt, MeasureTheory.charFun_dirac, HurwitzZeta.sinKernel_def, DirichletCharacter.LFunctionTrivChar_isBigO_near_one_horizontal, intervalIntegral.intervalIntegrable_cpow', arg_ofReal_of_neg, norm_cpow_eq_rpow_re_of_nonneg, fourier_coe_apply', IsExpCmpFilter.isLittleO_cpow_exp, cfc_real_eq_complex, TemperedDistribution.lineDerivOp_fourierInv_eq, HurwitzZeta.hurwitzOddFEPair_g, ProbabilityTheory.IsGaussian.charFunDual_eq_of_integral_eq_zero, norm_exp_mul_exp_add_exp_neg_le_of_abs_im_le, ofReal_im, ProbabilityTheory.IsGaussian.charFun_eq, integral_gaussian_complex, polarCoord_symm_apply, Real.hasFDerivAt_fourierChar_neg_bilinear_right, summable_bernoulli_fourier, WeakFEPair.f_modif_aux1, ofReal_ratCast, real_le_real, ZMod.toCircle_apply, GaussianFourier.integrable_cexp_neg_sum_mul_add, integral_exp_mul_complex_Iic, circleIntegral.integral_sub_center_inv, isAlgebraic_tan_rat_mul_pi, completedRiemannZeta_one, SchwartzMap.fourier_fderivCLM_eq, Gammaℝ_div_Gammaℝ_one_sub, Real.iteratedDeriv_fourierIntegral, cos_eq_neg_one_iff, Gamma_mul_Gamma_one_sub, log_neg_I, log_mul_ofReal, betaIntegral_convergent_left, jacobiTheta₂_add_left', hasSum_one_div_pow_mul_fourier_mul_bernoulliFun, re_add_im, conj_ofReal, norm_exp_ofReal_mul_I, Circle.coe_exp, real_lt_real, exp_two_pi_mul_I, isTheta_ofReal_left, ArithmeticFunction.LSeriesSummable_vonMangoldt, tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero, hasMellin_cpow_Ioc, integral_boundary_rect_of_differentiableOn_real, integral_mul_cpow_one_add_sq, IsExpCmpFilter.isLittleO_pow_mul_exp, HadamardThreeLines.scale_id_mem_verticalStrip_of_mem_verticalStrip, ofReal_cosh, ofReal_tsum, integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn, SlashInvariantForm.slash_action_eqn', isTheta_deriv_ofReal_cpow_const_atTop, neg_ofReal_mem_slitPlane, ProbabilityTheory.re_complexMGF_ofReal', ofReal_comp_neg, Real.hasDerivAt_fourierIntegral, cos_nat_mul_two_pi_sub, fourierCoeffOn_of_hasDerivAt, ofReal_ofNat, integrable_cexp_quadratic, image_exp_Ioc_eq_sphere, measurable_ofReal, ofRealAm_coe, ArithmeticFunction.iteratedDeriv_LSeries_alternating, two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable, exp_eq_one_iff, cos_nat_mul_two_pi_add_pi, monotone_ofReal, inv_Gammaℝ_one_sub, DifferentiableOn.circleIntegral_one_div_sub_center_pow_smul, exp_mul_I_antiperiodic, isLittleO_ofReal_right, ofReal_eq_re_of_isSelfAdjoint, isAlgebraic_cos_rat_mul_pi, HurwitzZeta.hasSum_int_completedCosZeta, TorusIntegrable.function_integrable, DirichletCharacter.LFunction_isBigO_horizontal, HurwitzZeta.LSeriesHasSum_sin, isometryOfOrthonormal_symm_apply, arg_mul_cos_add_sin_mul_I_sub, integrable_mul_cexp_neg_mul_sq, GammaIntegral_convergent, cos_two_pi_sub, ofReal_nnqsmul, ProbabilityTheory.IsGaussian.charFunDual_eq', enorm_exp_ofReal_mul_I, riemannZeta_two_mul_nat, Real.probChar_apply, Unitary.argSelfAdjoint_coe, ofReal_pow, ModularForm.logDeriv_qParam, tendsto_euler_sin_prod', ProbabilityTheory.charFun_gaussianReal, cosh_ofReal_re, HurwitzZeta.hurwitzZetaOdd_one_sub, HurwitzZeta.hasSum_int_completedSinZeta, Real.fourierInv_eq', cos_ofReal_im, ContDiff.real_of_complex, MeasureTheory.charFun_zero, MellinConvergent.cpow_smul, HurwitzZeta.hasSum_nat_hurwitzZetaOdd_of_mem_Icc, Real.fourierIntegral_deriv, norm_exp_I_mul_ofReal, tan_periodic, nnnorm_real, integrable_cexp_quadratic', sin_add_nat_mul_two_pi, norm_cpow_eq_rpow_re_of_pos, exp_mul_I_periodic, HurwitzZeta.hasSum_int_completedHurwitzZetaEven, log_ofReal_re, WeakFEPair.f_modif_aux2, mellin_cpow_smul, ofReal_inv, cos_eq_cos_iff, integral_cpow_mul_exp_neg_mul_Ioi, range_exp_mul_I, MeasureTheory.charFun_map_const_add, integral_cos_mul_complex, IsExpCmpFilter.isLittleO_cpow_mul_exp, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, summable_pow_mul_cexp, HurwitzZeta.sinZeta_one_sub, ModularFormClass.qExpansion_coeff_eq_intervalIntegral, approx_Gamma_integral_tendsto_Gamma_integral, ofReal_def, ProbabilityTheory.HasGaussianLaw.charFunDual_map_eq, summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp, inv_Gammaℝ_two_sub, Gamma_ofReal, Real.fderiv_fourierChar_neg_bilinear_left_apply, Gamma_one_half_eq, Real.fourier_eq', cos_int_mul_two_pi, WeakFEPair.Λ_residue_k, HadamardThreeLines.scale_id_mem_verticalClosedStrip_of_mem_verticalClosedStrip, integrable_cexp_neg_mul_sq, cos_two_pi, cos_add_int_mul_two_pi, fourierCoeffOn_of_hasDeriv_right, Real.rpow_def, ZetaAsymptotics.tendsto_riemannZeta_sub_one_div_nhds_right, normSq_ofReal_sub_I_mul_sqrt_one_sub, exp_int_mul_two_pi_mul_I, arg_cos_add_sin_mul_I_eq_toIocMod, continuousAt_ofReal_cpow_const, LSeries_eq_mul_integral', integrableOn_Ioi_deriv_norm_ofReal_cpow, isLittleO_ofReal_left, log_exp_exists, horizontalSegment_eq, ofReal_mul, fourier_gaussian_pi, continuousAt_gaussian_integral, Real.deriv_fourier, cfcₙ_complex_eq_real, PositiveLinearMap.preGNS_norm_sq, EulerSine.integral_cos_mul_cos_pow_even, GaussianInt.intCast_complex_norm, Gammaℂ_one, ofReal_choose, cos_pi, ModularGroup.smul_eq_lcRow0_add, ofReal_tanh_ofReal_re, fourierIntegral_gaussian_pi', tendsto_tsum_powerSeries_nhdsWithin_lt, LSeries.positive_of_differentiable_of_eqOn, integral_gaussian_complex_Ioi, ModularForm.constℝ_toFun, AnalyticOnNhd.eq_re_add_const_mul_I_of_re_eq_const, MeasureTheory.charFun_map_add_const, mellin_comp_mul_left, cpow_ofReal, sin_add_pi, differentiableAt_iteratedDerivWithin_cexp, pi_mul_cot_pi_q_exp, BoundedContinuousFunction.probCharDual_apply, hasDerivAt_Gamma_nat, cos_sub_int_mul_two_pi, sin_add_pi_div_two, mellin_comp_mul_right, NumberField.mixedEmbedding.normAtPlace_real, ofReal_tan, ModularForm.slash_apply, integrableOn_exp_mul_complex_Ioi, angle_exp_exp, re_ofReal_mul, lim_eq_lim_im_add_lim_re, rectangle_eq_convexHull, NumberField.mixedEmbedding.mixedSpaceOfRealSpace_apply, ProbabilityTheory.re_complexMGF_ofReal, UpperHalfPlane.petersson_slash, EisensteinSeries.tsum_symmetricIco_tsum_sub_eq
ofRealHom 📖	CompOp	3 math math: ringHom_eq_ofReal_of_continuous, ofRealHom_eq_coe, subfield_eq_of_closed
re 📖	CompOp	348 math math: mul_im, re_mul_ofReal, div_ofReal_re, isOpen_re_lt_EReal, coe_realPart, log_re, mellin_eq_fourier, Quaternion.coeComplex_re, AddDissociated.randomisation, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I, NumberField.mixedEmbedding.convexBodyLT'_mem, norm_sub_one_sq_eq_of_norm_eq_one, ProbabilityTheory.eqOn_complexMGF_of_mgf', re_eq_add_conj, HadamardThreeLines.norm_le_sSupNormIm, eq_re_of_ofReal_le, comap_exp_cobounded, norm_le_abs_re_add_abs_im, isTheta_cpow_rpow, UpperHalfPlane.re_smul, natCast_re, exp_re, AnalyticOnNhd.re_ofReal, not_le_iff, cos_ofReal_re, re_sqrt_ofReal, ofReal_cosh_ofReal_re, frontier_setOf_re_le, mul_re, reLm_coe, re_balance, equivRealProd_apply, IsExpCmpFilter.isBigO_im_pow_re, range_re, LSeriesSummable.abscissaOfAbsConv_le, norm_cexp_neg_mul_sq, ofReal_sin_ofReal_re, cfc_complex_eq_real, toMatrix_rotation, neg_pi_div_two_lt_arg_iff, arg_le_pi_div_two_iff, GaussianInt.toComplex_div_re, measurableEquivPi_apply, tendsto_exp_nhds_zero_iff, frontier_preimage_re, nndist_of_im_eq, log_exp_eq_re_add_toIocMod, tan_arg, interior_setOf_le_re, PositiveLinearMap.preGNS_norm_def, uniformContinuous_re, liftAux_apply, integrableOn_Ioi_cpow_iff, LSeries_deriv_eqOn, deriv_norm_ofReal_cpow, measurableEquivRealProd_apply, LSeries_analyticOnNhd, HasStrictDerivAt.real_of_complex, ofReal_mul', arg_of_im_pos, tan_ofReal_re, tendsto_exp_comap_re_atTop, convex_halfSpace_re_le, continuous_re, exp_ofReal_re, ofReal_cos_ofReal_re, integrableOn_Ioi_norm_cpow_iff, PhragmenLindelof.eqOn_quadrant_II, sub_re, sq_norm_sub_sq_re, comap_exp_nhds_zero, abs_re_eq_norm, norm_cpow_of_ne_zero, interior_preimage_re, PhragmenLindelof.eq_zero_on_quadrant_IV, norm_ofReal_cpow_eventually_eq_atTop, norm_mul_cos_arg, measurable_re, ProbabilityTheory.eqOn_complexMGF_of_mgf, div_ofNat, smul_re, uniformlyContinuous_re, map_exp_comap_re_atTop, orthonormalBasisOneI_repr_apply, tendsto_sub_mul_tsum_nat_cpow, norm_cpow_of_imp, re_comp_balance, exp_ofReal_mul_I_re, re_le_re, div_ratCast_re, IsSelfAdjoint.val_re_map_spectrum, PhragmenLindelof.eqOn_quadrant_I, normSq_add, cpow_ofReal_re, re_sq_le_normSq, div_im, PhragmenLindelof.eqOn_quadrant_III, LinearIsometry.re_apply_eq_re_of_add_conj_eq, sq_norm_sub_sq_im, LSeriesSummable.le_const_mul_rpow, arg_neg_eq_arg_add_pi_iff, ofReal_sinh_ofReal_re, abs_im_eq_norm, ratCast_re, isBigO_re_sub_re, comap_exp_nhdsNE, isBigO_cpow_rpow, eta, HadamardThreeLines.norm_invInterpStrip, cfcₙ_real_eq_complex, add_re, ofReal_cot_ofReal_re, normSq_sub, inv_re, IsExpCmpFilter.tendsto_re, intervalIntegral.integrableOn_Ioo_cpow_iff, arg_lt_pi_div_two_iff, map_exp_comap_re_atBot, intCast_re, sqrt_of_nonneg, equivRealProdAddHom_symm_apply_re, ProbabilityTheory.beta_eq_betaIntegralReal, selfAdjointEquiv_apply, UniformContinuousOn.cexp, neg_iff, equivRealProdLm_apply, div_intCast, inner_map_complex, arg_eq_nhds_of_im_neg, tendsto_exp_comap_re_atBot, nonpos_iff, arg_lt_pi_iff, arg_of_im_nonneg_of_ne_zero, dist_of_im_eq, NumberField.mixedEmbedding.exists_ne_zero_mem_ringOfIntegers_lt', arg_eq_pi_div_two_iff, UnitDisc.re_coe, SpectrumRestricts.real_iff, GaussianInt.to_real_re, LSeries.norm_term_eq, ProbabilityTheory.norm_complexMGF_le_mgf, conj_eq_iff_re, slitPlane_eq_union, mul_I_re, I_re, closure_preimage_re, ContDiffAt.real_of_complex, ext_iff, QuasispectrumRestricts.real_iff, unitary.two_mul_one_sub_le_norm_sub_one_sq, NumberField.mixedEmbedding.stdBasis_apply_isComplex_fst, add_conj, coe_basisOneI_repr, Polynomial.quadratic_dvd_of_aeval_eq_zero_im_ne_zero, AnalyticWithinAt.re_ofReal, GaussianInt.toComplex_re_div, tanh_ofReal_re, GaussianInt.toComplex_re, sinh_ofReal_re, cpow_inv_two_im_eq_sqrt, cpow_inv_two_re, div_re, abs_arg_le_pi_div_two_iff, sin_ofReal_re, mellin_eq_fourierIntegral, frontier_setOf_le_re, DirichletCharacter.LSeriesSummable_iff, ofReal_tan_ofReal_re, norm_natCast_cpow_of_pos, convex_halfSpace_re_ge, re_eq_ofReal_of_isSelfAdjoint, Algebra.leftMulMatrix_complex, Quaternion.re_coeComplex, frontier_setOf_re_lt, hasSum_re, MeasureTheory.SignedMeasure.toComplexMeasure_apply_re, re_ofNat, GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I', div_ofReal, convex_halfSpace_re_lt, ofReal_exp_ofReal_re, closure_setOf_re_lt, stolzCone_subset_stolzSet_aux, inner, AnalyticAt.harmonicAt_re, isHomeomorphicTrivialFiberBundle_re, RCLike.complexRingEquiv_symm_apply, selfAdjoint.mem_spectrum_eq_re, IsExpCmpFilter.isLittleO_log_norm_re, re_le_norm, equivRealProdAddHom_apply, interior_setOf_re_le, abs_re_le_norm, LSeries_differentiableOn, div_ofNat_re, LSeriesSummable_one_iff, equivRealProdLm_symm_apply_re, closure_setOf_lt_re, harmonic_is_realOfHolomorphic, re_qsmul, IsExpCmpFilter.isLittleO_im_pow_exp_re, re_tsum, TendstoUniformlyOn.re, div_intCast_re, isTheta_cpow_const_rpow, I_mul, arg_of_im_neg, I_mul_re, arg_eq_zero_iff, wedgeIntegral_add_wedgeIntegral_eq, norm_sub_one_sq_eqOn_sphere, abs_arg_lt_pi_div_two_iff, sqrt_eq_real_add_ite, not_le_zero_iff, pos_iff, NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt', sin_eq, Unitary.two_mul_one_sub_le_norm_sub_one_sq, ProbabilityTheory.differentiableOn_complexMGF, HadamardThreeLines.norm_lt_sSupNormIm_eps, re_eq_neg_norm, mellin_convergent_iff_norm, ArithmeticFunction.LSeriesSummable_moebius_iff, HasDerivAt.real_of_complex, PhragmenLindelof.eq_zero_on_quadrant_II, neg_re, Algebra.trace_complex_apply, conj_re, norm_exp, ArithmeticFunction.LSeriesSummable_zeta_iff, isBigO_deriv_ofReal_cpow_const_atTop, GaussianInt.re_toComplex, mem_slitPlane_iff, LSeriesSummable.isBigO_rpow, PhragmenLindelof.eqOn_quadrant_IV, riemannZeta_re_pos_of_one_lt, NumberField.mixedEmbedding.commMap_apply_of_isReal, LSeries_analyticOn, ArithmeticFunction.vonMangoldt.continuousOn_LFunctionResidueClassAux, equivRealProdCLM_symm_apply_re, cos_eq, sq_nonpos_iff, IsSelfAdjoint.mem_spectrum_eq_re, summable_one_div_nat_cpow, TendstoUniformly.re, zero_re, ofReal_re, isometryOfOrthonormal_apply, frontier_setOf_le_re_and_le_im, abs_re_lt_norm, not_lt_zero_iff, AnalyticAt.re_ofReal, norm_exp_eq_iff_re_eq, selfAdjoint.val_re_map_spectrum, IsExpCmpFilter.isLittleO_log_re_re, arg_eq_nhds_of_im_pos, ProbabilityTheory.analyticOn_complexMGF, re_surjective, RCLike.re_eq_complex_re, exp_eq_exp_re_mul_sin_add_cos, neg_pi_div_two_le_arg_iff, IsExpCmpFilter.abs_im_pow_eventuallyLE_exp_re, re_zsmul, I_mul_im, isQuotientMap_re, ArithmeticFunction.vonMangoldt.LFunctionResidueClassAux_real, cpow_inv_two_im_eq_neg_sqrt, imaginaryPart_smul, norm_natCast_cpow_le_norm_natCast_cpow_iff, norm_cpow_eq_rpow_re_of_nonneg, cfc_real_eq_complex, re_expect, norm_exp_mul_exp_add_exp_neg_le_of_abs_im_le, UpperHalfPlane.coe_re, re_nnqsmul, IsExpCmpFilter.tendsto_abs_re, HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip', re_add_im, nonneg_iff, normSq_apply, frontier_setOf_le_re_and_im_le, PhragmenLindelof.eq_zero_on_quadrant_I, realPart_smul, norm_eq_sqrt_sq_add_sq, isTheta_deriv_ofReal_cpow_const_atTop, ProbabilityTheory.re_complexMGF_ofReal', le_def, div_natCast, AnalyticOn.re_ofReal, norm_cpow_le, re_nsmul, edist_of_im_eq, ofReal_eq_re_of_isSelfAdjoint, Gamma_def, frontier_setOf_lt_re, convex_halfSpace_re_gt, one_re, ProbabilityTheory.analyticOnNhd_complexMGF, abs_cpow_inv_two_im, norm_le_sqrt_two_mul_max, cosh_ofReal_re, UpperHalfPlane.mk_re, ContDiff.real_of_complex, re_eq_norm, RCLike.re_to_complex, norm_cpow_eq_rpow_re_of_pos, re_sum, log_ofReal_re, norm_sub_one_sq_eq_of_norm_one, mem_reProdIm, hasSum_iff, ArithmeticFunction.vonMangoldt.eqOn_LFunctionResidueClassAux, isOpenMap_re, IsExpCmpFilter.isTheta_cpow_exp_re_mul_log, LinearIsometry.re_apply_eq_re, PhragmenLindelof.eq_zero_on_quadrant_III, isOpen_re_gt_EReal, neg_re_eq_norm, HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip₀₁', arg_neg_eq_arg_sub_pi_iff, mul_I_im, isCauSeq_re, re_nonneg_iff_nonneg, Real.rpow_def, arg_eq_pi_iff, div_natCast_re, re_nnratCast, exp_im, div_ratCast, cfcₙ_complex_eq_real, cos_arg, arg_eq_neg_pi_div_two_iff, norm_natCast_cpow_of_re_ne_zero, GaussianInt.intCast_re, coe_reAddGroupHom, equivRealProdCLM_apply, ofReal_tanh_ofReal_re, abs_re_div_norm_le_one, not_lt_iff, lim_re, tendsto_exp_comap_re_atBot_nhdsNE, dist_eq_re_im, lt_def, re_ofReal_mul, rectangle_eq_convexHull, reCLM_apply, ProbabilityTheory.re_complexMGF_ofReal, ModularGroup.coe_truncatedFundamentalDomain
reAddGroupHom 📖	CompOp	1 math math: coe_reAddGroupHom
reProdIm 📖	CompOp	16 math math: verticalSegment_eq, reProdIm_nonempty, reProdIm_subset_iff', Bornology.IsBounded.reProdIm, convexHull_reProdIm, reProdIm_subset_iff, reProdIm_eq_empty, IsOpen.reProdIm, preimage_equivRealProd_prod, interior_reProdIm, closure_reProdIm, IsCompact.reProdIm, frontier_reProdIm, mem_reProdIm, horizontalSegment_eq, IsClosed.reProdIm
«term_×ℂ_» 📖	CompOp	—

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
I_im 📖	mathematical	—	im I Real Real.instOne	—	—
I_mul 📖	mathematical	—	Complex instMul I Real Real.instNeg im re	—	ext_iff MulZeroClass.zero_mul one_mul zero_sub zero_add
I_mul_I 📖	mathematical	—	Complex instMul I instNeg instOne	—	ext_iff MulZeroClass.mul_zero mul_one zero_sub add_zero neg_zero
I_mul_im 📖	mathematical	—	im Complex instMul I re	—	MulZeroClass.zero_mul one_mul zero_add
I_mul_re 📖	mathematical	—	re Complex instMul I Real Real.instNeg im	—	MulZeroClass.zero_mul one_mul zero_sub
I_ne_zero 📖	—	—	—	—	zero_ne_one NeZero.charZero_one IsStrictOrderedRing.toCharZero Real.instIsStrictOrderedRing
I_pow_eq_pow_mod 📖	mathematical	—	Complex Monoid.toNatPow MonoidWithZero.toMonoid Semiring.toMonoidWithZero instSemiring I	—	pow_add pow_mul I_pow_four one_pow one_mul
I_pow_four 📖	mathematical	—	Complex Monoid.toNatPow MonoidWithZero.toMonoid Semiring.toMonoidWithZero instSemiring I instOne	—	pow_mul I_sq neg_one_sq
I_pow_three 📖	mathematical	—	Complex Monoid.toNatPow MonoidWithZero.toMonoid Semiring.toMonoidWithZero instSemiring I instNeg	—	pow_succ I_sq neg_one_mul
I_re 📖	mathematical	—	re I Real Real.instZero	—	—
I_sq 📖	mathematical	—	Complex Monoid.toNatPow MonoidWithZero.toMonoid Semiring.toMonoidWithZero instSemiring I instNeg instOne	—	sq I_mul_I
add_conj 📖	mathematical	—	Complex instAdd DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing ofReal Real Real.instMul instOfNatAtLeastTwo Real.instNatCast Nat.instAtLeastTwoHAddOfNat re	—	Nat.instAtLeastTwoHAddOfNat ext_iff two_mul add_neg_cancel
add_im 📖	mathematical	—	im Complex instAdd Real Real.instAdd	—	—
add_re 📖	mathematical	—	re Complex instAdd Real Real.instAdd	—	—
canLift 📖	mathematical	—	CanLift Complex Real ofReal im Real.instZero	—	ext
coe_imAddGroupHom 📖	mathematical	—	DFunLike.coe AddMonoidHom Complex Real AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddMonoidWithOne.toAddMonoid AddGroupWithOne.toAddMonoidWithOne addGroupWithOne Real.instAddMonoid AddMonoidHom.instFunLike imAddGroupHom im	—	—
coe_reAddGroupHom 📖	mathematical	—	DFunLike.coe AddMonoidHom Complex Real AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddMonoidWithOne.toAddMonoid AddGroupWithOne.toAddMonoidWithOne addGroupWithOne Real.instAddMonoid AddMonoidHom.instFunLike reAddGroupHom re	—	—
conj_I 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing I instNeg	—	ext_iff neg_zero
conj_eq_iff_im 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing im Real Real.instZero	—	add_self_eq_zero IsStrictOrderedRing.noZeroDivisors Real.instIsStrictOrderedRing AddGroup.existsAddOfLE IsStrictOrderedRing.toCharZero neg_eq_iff_add_eq_zero ext
conj_eq_iff_re 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing ofReal re	—	conj_eq_iff_real
conj_eq_iff_real 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing ofReal	—	ext eq_zero_of_neg_eq Real.instIsOrderedAddMonoid conj_ofReal
conj_im 📖	mathematical	—	im DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing Real Real.instNeg	—	—
conj_inv 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing instInv	—	star_inv₀
conj_natCast 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing instNatCast	—	map_natCast RingHom.instRingHomClass
conj_neg_I 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing instNeg I	—	map_neg RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass conj_I neg_neg
conj_ofNat 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing	—	map_ofNat RingHom.instRingHomClass
conj_ofReal 📖	mathematical	—	DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing ofReal	—	ext_iff neg_zero
conj_re 📖	mathematical	—	re DFunLike.coe RingHom Complex Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing	—	—
div_I 📖	mathematical	—	Complex DivInvMonoid.toDiv instDivInvMonoid I instNeg instMul	—	div_eq_iff_mul_eq I_ne_zero neg_mul mul_assoc I_mul_I mul_neg mul_one neg_neg
div_im 📖	mathematical	—	im Complex DivInvMonoid.toDiv instDivInvMonoid Real Real.instSub Real.instDivInvMonoid Real.instMul re DFunLike.coe MonoidWithZeroHom NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	div_eq_mul_inv inv_im neg_mul mul_neg inv_re add_comm mul_assoc sub_eq_add_neg
div_intCast 📖	mathematical	—	Complex DivInvMonoid.toDiv instDivInvMonoid instIntCast Real Real.instDivInvMonoid re Real.instIntCast im	—	div_ofReal
div_intCast_im 📖	mathematical	—	im Complex DivInvMonoid.toDiv instDivInvMonoid instIntCast Real Real.instDivInvMonoid Real.instIntCast	—	div_intCast
div_intCast_re 📖	mathematical	—	re Complex DivInvMonoid.toDiv instDivInvMonoid instIntCast Real Real.instDivInvMonoid Real.instIntCast	—	div_intCast
div_natCast 📖	mathematical	—	Complex DivInvMonoid.toDiv instDivInvMonoid instNatCast Real Real.instDivInvMonoid re Real.instNatCast im	—	div_ofReal
div_natCast_im 📖	mathematical	—	im Complex DivInvMonoid.toDiv instDivInvMonoid instNatCast Real Real.instDivInvMonoid Real.instNatCast	—	div_natCast
div_natCast_re 📖	mathematical	—	re Complex DivInvMonoid.toDiv instDivInvMonoid instNatCast Real Real.instDivInvMonoid Real.instNatCast	—	div_natCast
div_ofNat 📖	mathematical	—	Complex DivInvMonoid.toDiv instDivInvMonoid Real Real.instDivInvMonoid re im	—	div_natCast
div_ofNat_im 📖	mathematical	—	im Complex DivInvMonoid.toDiv instDivInvMonoid Real Real.instDivInvMonoid	—	div_natCast_im
div_ofNat_re 📖	mathematical	—	re Complex DivInvMonoid.toDiv instDivInvMonoid Real Real.instDivInvMonoid	—	div_natCast_re
div_ofReal 📖	mathematical	—	Complex DivInvMonoid.toDiv instDivInvMonoid ofReal Real Real.instDivInvMonoid re im	—	div_eq_inv_mul ofReal_mul'
div_ofReal_im 📖	mathematical	—	im Complex DivInvMonoid.toDiv instDivInvMonoid ofReal Real Real.instDivInvMonoid	—	div_ofReal
div_ofReal_re 📖	mathematical	—	re Complex DivInvMonoid.toDiv instDivInvMonoid ofReal Real Real.instDivInvMonoid	—	div_ofReal
div_ratCast 📖	mathematical	—	Complex DivInvMonoid.toDiv instDivInvMonoid instRatCast Real Real.instDivInvMonoid re Real.instRatCast im	—	div_ofReal
div_ratCast_im 📖	mathematical	—	im Complex DivInvMonoid.toDiv instDivInvMonoid instRatCast Real Real.instDivInvMonoid Real.instRatCast	—	div_ratCast
div_ratCast_re 📖	mathematical	—	re Complex DivInvMonoid.toDiv instDivInvMonoid instRatCast Real Real.instDivInvMonoid Real.instRatCast	—	div_ratCast
div_re 📖	mathematical	—	re Complex DivInvMonoid.toDiv instDivInvMonoid Real Real.instAdd Real.instDivInvMonoid Real.instMul DFunLike.coe MonoidWithZeroHom NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq im	—	div_eq_mul_inv inv_re inv_im neg_mul mul_neg sub_eq_add_neg neg_neg mul_assoc
equivRealProdAddHom_apply 📖	mathematical	—	DFunLike.coe AddEquiv Complex Real instAdd Prod.instAdd Real.instAdd EquivLike.toFunLike AddEquiv.instEquivLike equivRealProdAddHom re im	—	—
equivRealProdAddHom_symm_apply 📖	mathematical	—	DFunLike.coe AddEquiv Real Complex Prod.instAdd Real.instAdd instAdd EquivLike.toFunLike AddEquiv.instEquivLike AddEquiv.symm equivRealProdAddHom ofReal instMul I	—	equivRealProd_symm_apply
equivRealProdAddHom_symm_apply_im 📖	mathematical	—	im DFunLike.coe AddEquiv Real Complex Prod.instAdd Real.instAdd instAdd EquivLike.toFunLike AddEquiv.instEquivLike AddEquiv.symm equivRealProdAddHom	—	—
equivRealProdAddHom_symm_apply_re 📖	mathematical	—	re DFunLike.coe AddEquiv Real Complex Prod.instAdd Real.instAdd instAdd EquivLike.toFunLike AddEquiv.instEquivLike AddEquiv.symm equivRealProdAddHom	—	—
equivRealProd_apply 📖	mathematical	—	DFunLike.coe Equiv Complex Real EquivLike.toFunLike Equiv.instEquivLike equivRealProd re im	—	—
equivRealProd_symm_apply 📖	mathematical	—	DFunLike.coe Equiv Real Complex EquivLike.toFunLike Equiv.instEquivLike Equiv.symm equivRealProd instAdd ofReal instMul I	—	ext MulZeroClass.mul_zero mul_one sub_self add_zero zero_add
eta 📖	mathematical	—	re im	—	—
exists 📖	—	—	—	—	—
ext 📖	—	re im	—	—	—
ext_iff 📖	mathematical	—	re im	—	ext
forall 📖	—	—	—	—	—
horizontalSegment_eq 📖	mathematical	—	Set.image Real Complex instAdd ofReal instMul I Set.uIcc Real.lattice reProdIm Set Set.instSingletonSet	—	preimage_equivRealProd_prod Set.ext MulZeroClass.mul_zero mul_one sub_self add_zero zero_add re_add_im
im_eq_sub_conj 📖	mathematical	—	ofReal im Complex DivInvMonoid.toDiv instDivInvMonoid instSub DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing instMul instOfNatAtLeastTwo instNatCast Nat.instAtLeastTwoHAddOfNat I	—	Nat.instAtLeastTwoHAddOfNat sub_conj ofReal_mul mul_right_comm mul_div_cancel_left₀ GroupWithZero.toMulDivCancelClass mul_ne_zero IsDomain.to_noZeroDivisors Field.isDomain two_ne_zero CharZero.NeZero.two instCharZero I_ne_zero
im_mul_ofReal 📖	mathematical	—	im Complex instMul ofReal Real Real.instMul	—	MulZeroClass.mul_zero zero_add
im_nnqsmul 📖	mathematical	—	im NNRat Complex SMul.instSMulRealComplex SMulZeroClass.toSMul Real AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid DistribSMul.toSMulZeroClass NNRat.instDistribSMul Semifield.toDivisionSemiring Field.toSemifield Real.instField	—	smul_im
im_nnratCast 📖	mathematical	—	im NNRat.cast Complex instNNRatCast Real Real.instZero	—	—
im_nsmul 📖	mathematical	—	im Complex AddMonoid.toNatSMul AddMonoidWithOne.toAddMonoid AddGroupWithOne.toAddMonoidWithOne addGroupWithOne Real Real.instAddMonoid	—	smul_im
im_ofNat 📖	mathematical	—	im Real Real.instZero	—	—
im_ofReal_mul 📖	mathematical	—	im Complex instMul ofReal Real Real.instMul	—	MulZeroClass.zero_mul add_zero
im_qsmul 📖	mathematical	—	im Complex SMul.instSMulRealComplex SMulZeroClass.toSMul Real AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid DistribSMul.toSMulZeroClass Rat.instDistribSMul Real.instDivisionRing	—	smul_im
im_sq_le_normSq 📖	mathematical	—	Real Real.instLE Real.instMul im DFunLike.coe MonoidWithZeroHom Complex NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	le_add_of_nonneg_left covariant_swap_add_of_covariant_add IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid mul_self_nonneg AddGroup.existsAddOfLE IsOrderedRing.toPosMulMono Real.instIsOrderedRing
im_surjective 📖	mathematical	—	Complex Real im	—	—
im_zsmul 📖	mathematical	—	im Complex SubNegMonoid.toZSMul AddGroup.toSubNegMonoid AddGroupWithOne.toAddGroup addGroupWithOne Real Real.instAddGroup	—	smul_im
instCharZero 📖	mathematical	—	CharZero Complex AddGroupWithOne.toAddMonoidWithOne addGroupWithOne	—	charZero_of_inj_zero Nat.cast_eq_zero IsStrictOrderedRing.toCharZero Real.instIsStrictOrderedRing ofReal_eq_zero ofReal_natCast
instIsAddTorsionFree 📖	mathematical	—	IsAddTorsionFree Complex AddMonoidWithOne.toAddMonoid AddGroupWithOne.toAddMonoidWithOne addGroupWithOne	—	IsDomain.instIsAddTorsionFreeOfCharZero Field.isDomain instCharZero
instNontrivial 📖	mathematical	—	Nontrivial Complex	—	domain_nontrivial Real.instNontrivial
intCast_im 📖	mathematical	—	im Complex instIntCast Real Real.instZero	—	—
intCast_re 📖	mathematical	—	re Complex instIntCast Real Real.instIntCast	—	—
inv_I 📖	mathematical	—	Complex instInv I instNeg	—	inv_eq_one_div div_I one_mul
inv_def 📖	mathematical	—	Complex instInv instMul DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing ofReal Real Real.instInv MonoidWithZeroHom NonAssocSemiring.toMulZeroOneClass instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	—
inv_im 📖	mathematical	—	im Complex instInv Real DivInvMonoid.toDiv Real.instDivInvMonoid Real.instNeg DFunLike.coe MonoidWithZeroHom NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	MulZeroClass.mul_zero neg_mul zero_add division_def
inv_re 📖	mathematical	—	re Complex instInv Real DivInvMonoid.toDiv Real.instDivInvMonoid DFunLike.coe MonoidWithZeroHom NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	MulZeroClass.mul_zero sub_zero division_def
inv_zero 📖	mathematical	—	Complex instInv instZero	—	ofReal_zero ofReal_inv inv_zero
mem_reProdIm 📖	mathematical	—	Complex Set Set.instMembership reProdIm Real re im	—	—
mk_eq_add_mul_I 📖	mathematical	—	Complex instAdd ofReal instMul I	—	ext_iff MulZeroClass.mul_zero mul_one sub_self add_zero zero_add
mul_I_im 📖	mathematical	—	im Complex instMul I re	—	mul_one MulZeroClass.mul_zero add_zero
mul_I_re 📖	mathematical	—	re Complex instMul I Real Real.instNeg im	—	MulZeroClass.mul_zero mul_one zero_sub
mul_conj 📖	mathematical	—	Complex instMul DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing ofReal MonoidWithZeroHom Real NonAssocSemiring.toMulZeroOneClass instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	ext_iff mul_neg sub_eq_neg_add neg_neg add_comm mul_comm add_neg_cancel
mul_im 📖	mathematical	—	im Complex instMul Real Real.instAdd Real.instMul re	—	—
mul_inv_cancel 📖	mathematical	—	Complex instMul instInv instOne	—	inv_def mul_assoc mul_conj ofReal_mul mul_inv_cancel₀ normSq_eq_zero ofReal_one
mul_re 📖	mathematical	—	re Complex instMul Real Real.instSub Real.instMul im	—	—
natCast_im 📖	mathematical	—	im Complex instNatCast Real Real.instZero	—	—
natCast_re 📖	mathematical	—	re Complex instNatCast Real Real.instNatCast	—	—
neg_im 📖	mathematical	—	im Complex instNeg Real Real.instNeg	—	—
neg_re 📖	mathematical	—	re Complex instNeg Real Real.instNeg	—	—
normSq_I 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq I Real.instOne	—	MulZeroClass.mul_zero mul_one zero_add
normSq_add 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instAdd Real.instAdd Real.instMul instOfNatAtLeastTwo Real.instNatCast Nat.instAtLeastTwoHAddOfNat re instMul RingHom CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing	—	Nat.instAtLeastTwoHAddOfNat Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.mul_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.add_pf_add_lt Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pp_pf_overlap Mathlib.Meta.NormNum.IsNat.to_raw_eq Mathlib.Meta.NormNum.isNat_add Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.one_mul Mathlib.Tactic.Ring.mul_pf_left Mathlib.Tactic.Ring.mul_pf_right Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.add_pf_add_gt Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.add_pf_add_overlap Mathlib.Tactic.Ring.add_overlap_pf Mathlib.Tactic.Ring.cast_pos Mathlib.Meta.NormNum.isNat_ofNat Mathlib.Meta.NormNum.instAtLeastTwo Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.neg_congr Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.sub_pf Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Tactic.Ring.mul_one
normSq_add_mul_I 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instAdd ofReal instMul I Real.instAdd Monoid.toNatPow Real.instMonoid	—	mk_eq_add_mul_I sq
normSq_apply 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq Real.instAdd Real.instMul re im	—	—
normSq_conj 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq RingHom CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing	—	mul_neg neg_mul neg_neg
normSq_div 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq DivInvMonoid.toDiv instDivInvMonoid Real.instDivInvMonoid	—	map_div₀ MonoidWithZeroHom.monoidWithZeroHomClass
normSq_eq_conj_mul_self 📖	mathematical	—	ofReal DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instMul RingHom CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing	—	ext mul_comm mul_neg sub_neg_eq_add neg_add_cancel
normSq_eq_zero 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq Real.instZero instZero	—	ext eq_zero_of_mul_self_add_mul_self_eq_zero IsStrictOrderedRing.noZeroDivisors Real.instIsStrictOrderedRing AddGroup.existsAddOfLE IsOrderedRing.toPosMulMono Real.instIsOrderedRing IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid add_comm normSq_zero
normSq_intCast 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instIntCast Real.instMul Real.instIntCast	—	normSq_ofReal
normSq_inv 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instInv Real.instInv	—	map_inv₀ MonoidWithZeroHom.monoidWithZeroHomClass
normSq_mk 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq Real.instAdd Real.instMul	—	—
normSq_mul 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instMul Real.instMul	—	MonoidWithZeroHom.map_mul
normSq_natCast 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instNatCast Real.instMul Real.instNatCast	—	normSq_ofReal
normSq_neg 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instNeg	—	mul_neg neg_mul neg_neg
normSq_nonneg 📖	mathematical	—	Real Real.instLE Real.instZero DFunLike.coe MonoidWithZeroHom Complex NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	add_nonneg IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid mul_self_nonneg AddGroup.existsAddOfLE IsOrderedRing.toPosMulMono Real.instIsOrderedRing
normSq_ofNat 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq Real.instMul	—	normSq_natCast
normSq_ofReal 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq ofReal Real.instMul	—	MulZeroClass.mul_zero add_zero
normSq_one 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instOne Real.instOne	—	map_one MonoidHomClass.toOneHomClass MonoidWithZeroHomClass.toMonoidHomClass MonoidWithZeroHom.monoidWithZeroHomClass
normSq_pos 📖	mathematical	—	Real Real.instLT Real.instZero DFunLike.coe MonoidWithZeroHom Complex NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	LE.le.lt_iff_ne normSq_nonneg normSq_eq_zero
normSq_ratCast 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instRatCast Real.instMul Real.instRatCast	—	normSq_ofReal
normSq_sub 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instSub Real.instSub Real.instAdd Real.instMul instOfNatAtLeastTwo Real.instNatCast Nat.instAtLeastTwoHAddOfNat re instMul RingHom CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing	—	Nat.instAtLeastTwoHAddOfNat sub_eq_add_neg normSq_add normSq_neg map_neg RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass mul_neg Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.atom_pf Mathlib.Tactic.Ring.add_pf_add_lt Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Ring.neg_congr Mathlib.Tactic.Ring.mul_congr Mathlib.Tactic.Ring.cast_pos Mathlib.Meta.NormNum.isNat_ofNat Mathlib.Meta.NormNum.instAtLeastTwo Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_pf_right Mathlib.Tactic.Ring.mul_one Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_mul Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.sub_pf
normSq_zero 📖	mathematical	—	DFunLike.coe MonoidWithZeroHom Complex Real NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq instZero Real.instZero	—	map_zero MonoidWithZeroHomClass.toZeroHomClass MonoidWithZeroHom.monoidWithZeroHomClass
ofRealHom_eq_coe 📖	mathematical	—	DFunLike.coe RingHom Real Complex Semiring.toNonAssocSemiring Real.semiring instSemiring RingHom.instFunLike ofRealHom ofReal	—	—
ofReal_add 📖	mathematical	—	ofReal Real Real.instAdd Complex instAdd	—	ext_iff add_zero
ofReal_comp_add 📖	mathematical	—	Real Complex ofReal Pi.instAdd Real.instAdd instAdd	—	map_comp_add AddMonoidHomClass.toAddHomClass RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass
ofReal_comp_mul 📖	mathematical	—	Real Complex ofReal Pi.instMul Real.instMul instMul	—	map_comp_mul NonUnitalRingHomClass.toMulHomClass RingHomClass.toNonUnitalRingHomClass RingHom.instRingHomClass
ofReal_comp_neg 📖	mathematical	—	Real Complex ofReal Pi.instNeg Real.instNeg instNeg	—	map_comp_neg RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass
ofReal_comp_nsmul 📖	mathematical	—	Real Complex ofReal AddMonoid.toNatSMul Pi.addMonoid Real.instAddMonoid AddMonoidWithOne.toAddMonoid AddGroupWithOne.toAddMonoidWithOne addGroupWithOne	—	map_comp_nsmul RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass
ofReal_comp_pow 📖	mathematical	—	Real Complex ofReal Pi.instPow Monoid.toNatPow Real.instMonoid MonoidWithZero.toMonoid Semiring.toMonoidWithZero instSemiring	—	map_comp_pow MonoidWithZeroHomClass.toMonoidHomClass RingHomClass.toMonoidWithZeroHomClass RingHom.instRingHomClass
ofReal_comp_sub 📖	mathematical	—	Real Complex ofReal Pi.instSub Real.instSub instSub	—	map_comp_sub RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass
ofReal_comp_zsmul 📖	mathematical	—	Real Complex ofReal SubNegMonoid.toZSMul Pi.subNegMonoid AddGroup.toSubNegMonoid Real.instAddGroup AddGroupWithOne.toAddGroup addGroupWithOne	—	map_comp_zsmul RingHomClass.toAddMonoidHomClass RingHom.instRingHomClass
ofReal_def 📖	mathematical	—	ofReal Real Real.instZero	—	—
ofReal_div 📖	mathematical	—	ofReal Real DivInvMonoid.toDiv Real.instDivInvMonoid Complex instDivInvMonoid	—	map_div₀ RingHomClass.toMonoidWithZeroHomClass RingHom.instRingHomClass
ofReal_eq_one 📖	mathematical	—	ofReal Complex instOne Real Real.instOne	—	—
ofReal_eq_zero 📖	mathematical	—	ofReal Complex instZero Real Real.instZero	—	—
ofReal_im 📖	mathematical	—	im ofReal Real Real.instZero	—	—
ofReal_inj 📖	mathematical	—	ofReal	—	—
ofReal_injective 📖	mathematical	—	Real Complex ofReal	—	—
ofReal_intCast 📖	mathematical	—	ofReal Real Real.instIntCast Complex instIntCast	—	—
ofReal_inv 📖	mathematical	—	ofReal Real Real.instInv Complex instInv	—	ext_iff inv_re MulZeroClass.mul_zero add_zero div_self_mul_self' inv_im neg_zero zero_div
ofReal_mul 📖	mathematical	—	ofReal Real Real.instMul Complex instMul	—	ext_iff MulZeroClass.mul_zero sub_zero MulZeroClass.zero_mul add_zero
ofReal_mul' 📖	mathematical	—	Complex instMul ofReal Real Real.instMul re im	—	ext re_ofReal_mul im_ofReal_mul
ofReal_natCast 📖	mathematical	—	ofReal Real Real.instNatCast Complex instNatCast	—	—
ofReal_ne_one 📖	—	—	—	—	ofReal_eq_one
ofReal_ne_zero 📖	—	—	—	—	ofReal_eq_zero
ofReal_neg 📖	mathematical	—	ofReal Real Real.instNeg Complex instNeg	—	ext_iff neg_zero
ofReal_nnqsmul 📖	mathematical	—	ofReal NNRat Real SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid DistribSMul.toSMulZeroClass NNRat.instDistribSMul Semifield.toDivisionSemiring Field.toSemifield Real.instField Complex SMul.instSMulRealComplex	—	NNRat.smul_def ofReal_mul
ofReal_nnratCast 📖	mathematical	—	ofReal NNRat.cast Real Real.instNNRatCast Complex instNNRatCast	—	—
ofReal_nsmul 📖	mathematical	—	ofReal Real AddMonoid.toNatSMul Real.instAddMonoid Complex AddMonoidWithOne.toAddMonoid AddGroupWithOne.toAddMonoidWithOne addGroupWithOne	—	nsmul_eq_mul ofReal_mul
ofReal_ofNat 📖	mathematical	—	ofReal	—	—
ofReal_one 📖	mathematical	—	ofReal Real Real.instOne Complex instOne	—	—
ofReal_pow 📖	mathematical	—	ofReal Real Monoid.toNatPow Real.instMonoid Complex MonoidWithZero.toMonoid Semiring.toMonoidWithZero instSemiring	—	pow_zero pow_succ ofReal_mul
ofReal_qsmul 📖	mathematical	—	ofReal Real SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid DistribSMul.toSMulZeroClass Rat.instDistribSMul Real.instDivisionRing Complex SMul.instSMulRealComplex	—	Rat.smul_def ofReal_mul
ofReal_ratCast 📖	mathematical	—	ofReal Real Real.instRatCast Complex instRatCast	—	—
ofReal_re 📖	mathematical	—	re ofReal	—	—
ofReal_sub 📖	mathematical	—	ofReal Real Real.instSub Complex instSub	—	ext_iff sub_self
ofReal_zero 📖	mathematical	—	ofReal Real Real.instZero Complex instZero	—	—
ofReal_zpow 📖	mathematical	—	ofReal Real DivInvMonoid.toZPow Real.instDivInvMonoid Complex instDivInvMonoid	—	map_zpow₀ RingHomClass.toMonoidWithZeroHomClass RingHom.instRingHomClass
ofReal_zsmul 📖	mathematical	—	ofReal Real SubNegMonoid.toZSMul AddGroup.toSubNegMonoid Real.instAddGroup Complex AddGroupWithOne.toAddGroup addGroupWithOne	—	zsmul_eq_mul ofReal_mul
one_im 📖	mathematical	—	im Complex instOne Real Real.instZero	—	—
one_re 📖	mathematical	—	re Complex instOne Real Real.instOne	—	—
preimage_equivRealProd_prod 📖	mathematical	—	Set.preimage Complex Real DFunLike.coe Equiv EquivLike.toFunLike Equiv.instEquivLike equivRealProd SProd.sprod Set Set.instSProd reProdIm	—	—
range_im 📖	mathematical	—	Set.range Real Complex im Set.univ	—	Function.Surjective.range_eq im_surjective
range_re 📖	mathematical	—	Set.range Real Complex re Set.univ	—	Function.Surjective.range_eq re_surjective
ratCast_im 📖	mathematical	—	im Complex instRatCast Real Real.instZero	—	—
ratCast_re 📖	mathematical	—	re Complex instRatCast Real Real.instRatCast	—	—
reProdIm_eq_empty 📖	mathematical	—	reProdIm Set Complex Set.instEmptyCollection Real	—	—
reProdIm_nonempty 📖	mathematical	—	Set.Nonempty Complex reProdIm Real	—	—
reProdIm_subset_iff 📖	mathematical	—	Set Complex Set.instHasSubset reProdIm Real SProd.sprod Set.instSProd	—	preimage_equivRealProd_prod Equiv.preimage_subset
reProdIm_subset_iff' 📖	mathematical	—	Set Complex Set.instHasSubset reProdIm Real Set.instEmptyCollection	—	reProdIm_subset_iff Set.prod_subset_prod_iff
re_add_im 📖	mathematical	—	Complex instAdd ofReal re instMul im I	—	ext_iff MulZeroClass.mul_zero mul_one sub_self add_zero zero_add
re_eq_add_conj 📖	mathematical	—	ofReal re Complex DivInvMonoid.toDiv instDivInvMonoid instAdd DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing instOfNatAtLeastTwo instNatCast Nat.instAtLeastTwoHAddOfNat	—	Nat.instAtLeastTwoHAddOfNat add_conj ofReal_mul mul_div_cancel_left₀ GroupWithZero.toMulDivCancelClass two_ne_zero CharZero.NeZero.two instCharZero
re_mul_ofReal 📖	mathematical	—	re Complex instMul ofReal Real Real.instMul	—	MulZeroClass.mul_zero sub_zero
re_nnqsmul 📖	mathematical	—	re NNRat Complex SMul.instSMulRealComplex SMulZeroClass.toSMul Real AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid DistribSMul.toSMulZeroClass NNRat.instDistribSMul Semifield.toDivisionSemiring Field.toSemifield Real.instField	—	smul_re
re_nnratCast 📖	mathematical	—	re NNRat.cast Complex instNNRatCast Real Real.instNNRatCast	—	—
re_nsmul 📖	mathematical	—	re Complex AddMonoid.toNatSMul AddMonoidWithOne.toAddMonoid AddGroupWithOne.toAddMonoidWithOne addGroupWithOne Real Real.instAddMonoid	—	smul_re
re_ofNat 📖	mathematical	—	re	—	—
re_ofReal_mul 📖	mathematical	—	re Complex instMul ofReal Real Real.instMul	—	MulZeroClass.zero_mul sub_zero
re_qsmul 📖	mathematical	—	re Complex SMul.instSMulRealComplex SMulZeroClass.toSMul Real AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid DistribSMul.toSMulZeroClass Rat.instDistribSMul Real.instDivisionRing	—	smul_re
re_sq_le_normSq 📖	mathematical	—	Real Real.instLE Real.instMul re DFunLike.coe MonoidWithZeroHom Complex NonAssocSemiring.toMulZeroOneClass Semiring.toNonAssocSemiring instSemiring Real.semiring MonoidWithZeroHom.funLike normSq	—	le_add_of_nonneg_right IsOrderedAddMonoid.toAddLeftMono Real.instIsOrderedAddMonoid mul_self_nonneg AddGroup.existsAddOfLE IsOrderedRing.toPosMulMono Real.instIsOrderedRing
re_surjective 📖	mathematical	—	Complex Real re	—	—
re_zsmul 📖	mathematical	—	re Complex SubNegMonoid.toZSMul AddGroup.toSubNegMonoid AddGroupWithOne.toAddGroup addGroupWithOne Real Real.instAddGroup	—	smul_re
real_smul 📖	mathematical	—	Real Complex SMul.instSMulRealComplex SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass Real.instAddMonoid DistribSMul.toSMulZeroClass NonUnitalNonAssocSemiring.toDistribSMul NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing Real.commRing instMul ofReal	—	—
smul_im 📖	mathematical	—	im Complex SMul.instSMulRealComplex Real	—	MulZeroClass.zero_mul sub_zero add_zero
smul_re 📖	mathematical	—	re Complex SMul.instSMulRealComplex Real	—	MulZeroClass.zero_mul sub_zero add_zero
star_def 📖	mathematical	—	Star.star Complex InvolutiveStar.toStar StarAddMonoid.toInvolutiveStar AddCommMonoid.toAddMonoid NonUnitalNonAssocSemiring.toAddCommMonoid NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing CommRing.toNonUnitalCommRing commRing StarRing.toStarAddMonoid instStarRing DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd	—	—
sub_conj 📖	mathematical	—	Complex instSub DFunLike.coe RingHom Semiring.toNonAssocSemiring CommSemiring.toSemiring instCommSemiring RingHom.instFunLike starRingEnd instStarRing instMul ofReal Real Real.instMul instOfNatAtLeastTwo Real.instNatCast Nat.instAtLeastTwoHAddOfNat im I	—	Nat.instAtLeastTwoHAddOfNat ext_iff sub_eq_add_neg add_neg_cancel two_mul MulZeroClass.mul_zero mul_one sub_self neg_neg add_zero
sub_im 📖	mathematical	—	im Complex instSub Real Real.instSub	—	—
sub_re 📖	mathematical	—	re Complex instSub Real Real.instSub	—	—
verticalSegment_eq 📖	mathematical	—	Set.image Real Complex instAdd ofReal instMul I Set.uIcc Real.lattice reProdIm Set Set.instSingletonSet	—	preimage_equivRealProd_prod Set.ext MulZeroClass.mul_zero mul_one sub_self add_zero zero_add Set.singleton_prod re_add_im
zero_im 📖	mathematical	—	im Complex instZero Real Real.instZero	—	—
zero_re 📖	mathematical	—	re Complex instZero Real Real.instZero	—	—

Complex.SMul

Definitions

Name	Category	Theorems
instSMulRealComplex 📖	CompOp	13 math math: Complex.im_qsmul, Complex.real_smul, Complex.instIsScalarTowerOfReal, Complex.im_nnqsmul, Complex.smul_re, Complex.instSMulCommClassOfReal, Complex.instStarModuleReal, Complex.instIsCentralScalarOfReal, Complex.re_qsmul, Complex.ofReal_qsmul, Complex.re_nnqsmul, Complex.ofReal_nnqsmul, Complex.smul_im

(root)

Definitions

Name	Category	Theorems
termℂ 📖	CompOp	—

---

← Back to Index