Locally uniform limits of holomorphic functions

This file gathers some results about locally uniform limits of holomorphic functions on an open subset of the complex plane.

Main results

• TendstoLocallyUniformlyOn.differentiableOn: A locally uniform limit of holomorphic functions is holomorphic.
• TendstoLocallyUniformlyOn.deriv: Locally uniform convergence implies locally uniform convergence of the derivatives to the derivative of the limit.

noncomputable def Complex.cderiv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (r : ℝ) (f : ℂ → E) (z : ℂ) : E

A circle integral which coincides with deriv f z whenever one can apply the Cauchy formula for the derivative. It is useful in the proof that locally uniform limits of holomorphic functions are holomorphic, because it depends continuously on f for the uniform topology.

theorem Complex.cderiv_eq_deriv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set ℂ} {z : ℂ} {r : ℝ} {f : ℂ → E} [CompleteSpace E] (hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (hr : 0 < r) (hzr : Metric.closedBall z r ⊆ U) : cderiv r f z = deriv f z

theorem Complex.norm_cderiv_le {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {M r : ℝ} {f : ℂ → E} (hr : 0 < r) (hf : ∀ w ∈ Metric.sphere z r, ‖f w‖ ≤ M) : ‖cderiv r f z‖ ≤ M / r

theorem Complex.cderiv_sub {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {r : ℝ} {f g : ℂ → E} (hr : 0 < r) (hf : ContinuousOn f (Metric.sphere z r)) (hg : ContinuousOn g (Metric.sphere z r)) : cderiv r (f - g) z = cderiv r f z - cderiv r g z

theorem Complex.norm_cderiv_lt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {M r : ℝ} {f : ℂ → E} (hr : 0 < r) (hfM : ∀ w ∈ Metric.sphere z r, ‖f w‖ < M) (hf : ContinuousOn f (Metric.sphere z r)) : ‖cderiv r f z‖ < M / r

theorem Complex.norm_cderiv_sub_lt {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {z : ℂ} {M r : ℝ} {f g : ℂ → E} (hr : 0 < r) (hfg : ∀ w ∈ Metric.sphere z r, ‖f w - g w‖ < M) (hf : ContinuousOn f (Metric.sphere z r)) (hg : ContinuousOn g (Metric.sphere z r)) : ‖cderiv r f z - cderiv r g z‖ < M / r

theorem TendstoUniformlyOn.cderiv {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {K : Set ℂ} {δ : ℝ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} (hF : TendstoUniformlyOn F f φ (Metric.cthickening δ K)) (hδ : 0 < δ) (hFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (Metric.cthickening δ K)) : TendstoUniformlyOn (Complex.cderiv δ ∘ F) (Complex.cderiv δ f) φ K

theorem Complex.tendstoUniformlyOn_deriv_of_cthickening_subset {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {U K : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} [CompleteSpace E] (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) {δ : ℝ} (hδ : 0 < δ) (hK : IsCompact K) (hU : IsOpen U) (hKU : Metric.cthickening δ K ⊆ U) : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K

theorem Complex.exists_cthickening_tendstoUniformlyOn {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {U K : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} [CompleteSpace E] (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) : ∃ δ > 0, Metric.cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K

theorem TendstoLocallyUniformlyOn.differentiableOn {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} [CompleteSpace E] [φ.NeBot] (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U) : DifferentiableOn ℂ f U

A locally uniform limit of holomorphic functions on an open domain of the complex plane is holomorphic (the derivatives converge locally uniformly to that of the limit, which is proved as TendstoLocallyUniformlyOn.deriv).

theorem TendstoLocallyUniformlyOn.deriv {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set ℂ} {φ : Filter ι} {F : ι → ℂ → E} {f : ℂ → E} [CompleteSpace E] (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U) : TendstoLocallyUniformlyOn (_root_.deriv ∘ F) (_root_.deriv f) φ U

theorem Complex.differentiableOn_tsum_of_summable_norm {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set ℂ} {F : ι → ℂ → E} [CompleteSpace E] {u : ι → ℝ} (hu : Summable u) (hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U) (hU : IsOpen U) (hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i) : DifferentiableOn ℂ (fun (w : ℂ) => ∑' (i : ι), F i w) U

If the terms in the sum ∑' (i : ι), F i are uniformly bounded on U by a summable function, and each term in the sum is differentiable on U, then so is the sum.

theorem Complex.hasSum_deriv_of_summable_norm {E : Type u_1} {ι : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {U : Set ℂ} {z : ℂ} {F : ι → ℂ → E} [CompleteSpace E] {u : ι → ℝ} (hu : Summable u) (hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U) (hU : IsOpen U) (hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i) (hz : z ∈ U) : HasSum (fun (i : ι) => deriv (F i) z) (deriv (fun (w : ℂ) => ∑' (i : ι), F i w) z)

If the terms in the sum ∑' (i : ι), F i are uniformly bounded on U by a summable function, then the sum of deriv F i at a point in U is the derivative of the sum.

theorem Complex.logDeriv_tendsto {ι : Type u_3} {p : Filter ι} {f : ι → ℂ → ℂ} {g : ℂ → ℂ} {s : Set ℂ} (hs : IsOpen s) {x : ℂ} (hx : x ∈ s) (hF : TendstoLocallyUniformlyOn f g p s) (hf : ∀ᶠ (n : ι) in p, DifferentiableOn ℂ (f n) s) (hg : g x ≠ 0) : Filter.Tendsto (fun (n : ι) => logDeriv (f n) x) p (nhds (logDeriv g x))

The logarithmic derivative of a sequence of functions converging locally uniformly to a function is the logarithmic derivative of the limit function.