Documentation

Mathlib.Analysis.SpecialFunctions.Complex.Circle

Maps on the unit circle #

In this file we prove some basic lemmas about Circle.exp and the restriction of Complex.arg to the unit circle. These two maps define a partial equivalence between Circle and ℝ, see Circle.argPartialEquiv and Circle.argEquiv, that sends the whole circle to (-π, π].

theorem Circle.injective_arg :

Function.Injective fun (z : Circle) => (↑z).arg

@[simp]

theorem Circle.arg_eq_arg {z w : Circle} :

(↑z).arg = (↑w).arg ↔ z = w

theorem Circle.arg_exp {x : ℝ} (h₁ : -Real.pi < x) (h₂ : x ≤ Real.pi) :

(↑(exp x)).arg = x

@[simp]

theorem Circle.exp_arg (z : Circle) :

exp (↑z).arg = z

noncomputable def Circle.argPartialEquiv :

PartialEquiv Circle ℝ

Complex.arg ∘ (↑) and Circle.exp define a partial equivalence between Circle and ℝ with source = Set.univ and target = Set.Ioc (-π) π.

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@[simp]

theorem Circle.argPartialEquiv_symm_apply :

↑argPartialEquiv.symm = ⇑exp

@[simp]

theorem Circle.argPartialEquiv_apply :

↑argPartialEquiv = Complex.arg ∘ Subtype.val

@[simp]

theorem Circle.argPartialEquiv_target :

argPartialEquiv.target = Set.Ioc (-Real.pi) Real.pi

@[simp]

theorem Circle.argPartialEquiv_source :

argPartialEquiv.source = Set.univ

noncomputable def Circle.argEquiv :

Circle ≃ ↑(Set.Ioc (-Real.pi) Real.pi)

Complex.arg and Circle.exp ∘ (↑) define an equivalence between Circle and (-π, π].

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@[simp]

theorem Circle.argEquiv_apply_coe (z : Circle) :

↑(argEquiv z) = (↑z).arg

@[simp]

theorem Circle.argEquiv_symm_apply :

⇑argEquiv.symm = ⇑exp ∘ Subtype.val

theorem Circle.leftInverse_exp_arg :

Function.LeftInverse (⇑exp) (Complex.arg ∘ Subtype.val)

theorem Circle.invOn_arg_exp :

Set.InvOn (Complex.arg ∘ Subtype.val) (⇑exp) (Set.Ioc (-Real.pi) Real.pi) Set.univ

theorem Circle.surjOn_exp_neg_pi_pi :

Set.SurjOn (⇑exp) (Set.Ioc (-Real.pi) Real.pi) Set.univ

theorem Circle.exp_eq_exp {x y : ℝ} :

exp x = exp y ↔ ∃ (m : ℤ), x = y + ↑m * (2 * Real.pi)

theorem Circle.periodic_exp :

Function.Periodic (⇑exp) (2 * Real.pi)

@[simp]

theorem Circle.exp_two_pi :

exp (2 * Real.pi) = 1

theorem Circle.exp_int_mul_two_pi (n : ℤ) :

exp (↑n * (2 * Real.pi)) = 1

theorem Circle.exp_two_pi_mul_int (n : ℤ) :

exp (2 * Real.pi * ↑n) = 1

theorem Circle.exp_eq_one {r : ℝ} :

exp r = 1 ↔ ∃ (n : ℤ), r = ↑n * (2 * Real.pi)

theorem Circle.exp_inj {r s : ℝ} :

exp r = exp s ↔ r ≡ s [PMOD 2 * Real.pi ]

theorem Circle.exp_sub_two_pi (x : ℝ) :

exp (x - 2 * Real.pi) = exp x

theorem Circle.exp_add_two_pi (x : ℝ) :

exp (x + 2 * Real.pi) = exp x

noncomputable def Real.Angle.toCircle (θ : Angle) :

Circle.exp, applied to a Real.Angle.

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@[simp]

theorem Real.Angle.toCircle_coe (x : ℝ) :

(↑x).toCircle = Circle.exp x

theorem Real.Angle.coe_toCircle (θ : Angle) :

↑θ.toCircle = ↑θ.cos + ↑θ.sin * Complex.I

@[simp]

theorem Real.Angle.toCircle_zero :

@[simp]

theorem Real.Angle.toCircle_neg (θ : Angle) :

(-θ).toCircle = θ.toCircle⁻¹

@[simp]

theorem Real.Angle.toCircle_add (θ₁ θ₂ : Angle) :

(θ₁ + θ₂).toCircle = θ₁.toCircle * θ₂.toCircle

@[simp]

theorem Real.Angle.arg_toCircle (θ : Angle) :

↑(↑θ.toCircle).arg = θ

Map from `AddCircle` to `Circle` #

theorem AddCircle.scaled_exp_map_periodic {T : ℝ} :

Function.Periodic (fun (x : ℝ) => Circle.exp (2 * Real.pi / T * x)) T

noncomputable def AddCircle.toCircle {T : ℝ} :

AddCircle T → Circle

The canonical map fun x => exp (2 π i x / T) from ℝ / ℤ • T to the unit circle in ℂ. If T = 0 we understand this as the constant function 1.

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theorem AddCircle.toCircle_apply_mk {T : ℝ} (x : ℝ) :

toCircle ↑x = Circle.exp (2 * Real.pi / T * x)

theorem AddCircle.toCircle_add {T : ℝ} (x y : AddCircle T) :

(x + y).toCircle = x.toCircle * y.toCircle

@[simp]

theorem AddCircle.toCircle_zero {T : ℝ} :

theorem AddCircle.toCircle_neg {T : ℝ} (x : AddCircle T) :

(-x).toCircle = x.toCircle⁻¹

theorem AddCircle.toCircle_nsmul {T : ℝ} (x : AddCircle T) (n : ℕ) :

(n • x).toCircle = x.toCircle ^ n

theorem AddCircle.toCircle_zsmul {T : ℝ} (x : AddCircle T) (n : ℤ) :

(n • x).toCircle = x.toCircle ^ n

theorem AddCircle.continuous_toCircle {T : ℝ} :

Continuous toCircle

theorem AddCircle.injective_toCircle {T : ℝ} (hT : T ≠ 0) :

Function.Injective toCircle

noncomputable def AddCircle.homeomorphCircle' :

AddCircle (2 * Real.pi) ≃ₜ Circle

The homeomorphism between AddCircle (2 * π) and Circle.

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@[simp]

theorem AddCircle.homeomorphCircle'_symm_apply (x : Circle) :

homeomorphCircle'.symm x = ↑(↑x).arg

@[simp]

theorem AddCircle.homeomorphCircle'_apply (θ : Real.Angle) :

homeomorphCircle' θ = θ.toCircle

theorem AddCircle.homeomorphCircle'_apply_mk (x : ℝ) :

homeomorphCircle' ↑x = Circle.exp x

noncomputable def AddCircle.homeomorphCircle {T : ℝ} (hT : T ≠ 0) :

AddCircle T ≃ₜ Circle

The homeomorphism between AddCircle and Circle.

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theorem AddCircle.homeomorphCircle_apply {T : ℝ} (hT : T ≠ 0) (x : AddCircle T) :

(homeomorphCircle hT) x = x.toCircle

theorem Circle.isAddQuotientCoveringMap_exp :

IsAddQuotientCoveringMap ⇑exp ↥(AddSubgroup.zmultiples (2 * Real.pi))

theorem Circle.isCoveringMap_exp :

IsCoveringMap ⇑exp

theorem isLocalHomeomorph_circleExp :

IsLocalHomeomorph ⇑Circle.exp

theorem Circle.isQuotientCoveringMap_zpow (n : ℤ) [NeZero n] :

IsQuotientCoveringMap (fun (x : Circle) => x ^ n) ↥(zpowGroupHom n).ker

theorem Circle.isQuotientCoveringMap_npow (n : ℕ) [NeZero n] :

IsQuotientCoveringMap (fun (x : Circle) => x ^ n) ↥(powMonoidHom n).ker