Locally uniform limits of holomorphic functions #
THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.
This file gathers some results about locally uniform limits of holomorphic functions on an open subset of the complex plane.
Main results #
tendsto_locally_uniformly_on.differentiable_on: A locally uniform limit of holomorphic functions is holomorphic.tendsto_locally_uniformly_on.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}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space 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}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{U : set ℂ}
{z : ℂ}
{r : ℝ}
{f : ℂ → E}
(hU : is_open U)
(hf : differentiable_on ℂ f U)
(hr : 0 < r)
(hzr : metric.closed_ball z r ⊆ U) :
complex.cderiv r f z = deriv f z
theorem
complex.norm_cderiv_le
{E : Type u_1}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{z : ℂ}
{M r : ℝ}
{f : ℂ → E}
(hr : 0 < r)
(hf : ∀ (w : ℂ), w ∈ metric.sphere z r → ‖f w‖ ≤ M) :
‖complex.cderiv r f z‖ ≤ M / r
theorem
complex.cderiv_sub
{E : Type u_1}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{z : ℂ}
{r : ℝ}
{f g : ℂ → E}
(hr : 0 < r)
(hf : continuous_on f (metric.sphere z r))
(hg : continuous_on g (metric.sphere z r)) :
complex.cderiv r (f - g) z = complex.cderiv r f z - complex.cderiv r g z
theorem
complex.norm_cderiv_lt
{E : Type u_1}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{z : ℂ}
{M r : ℝ}
{f : ℂ → E}
(hr : 0 < r)
(hfM : ∀ (w : ℂ), w ∈ metric.sphere z r → ‖f w‖ < M)
(hf : continuous_on f (metric.sphere z r)) :
‖complex.cderiv r f z‖ < M / r
theorem
complex.norm_cderiv_sub_lt
{E : Type u_1}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{z : ℂ}
{M r : ℝ}
{f g : ℂ → E}
(hr : 0 < r)
(hfg : ∀ (w : ℂ), w ∈ metric.sphere z r → ‖f w - g w‖ < M)
(hf : continuous_on f (metric.sphere z r))
(hg : continuous_on g (metric.sphere z r)) :
‖complex.cderiv r f z - complex.cderiv r g z‖ < M / r
theorem
complex.tendsto_uniformly_on.cderiv
{E : Type u_1}
{ι : Type u_2}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{K : set ℂ}
{δ : ℝ}
{φ : filter ι}
{F : ι → ℂ → E}
{f : ℂ → E}
(hF : tendsto_uniformly_on F f φ (metric.cthickening δ K))
(hδ : 0 < δ)
(hFn : ∀ᶠ (n : ι) in φ, continuous_on (F n) (metric.cthickening δ K)) :
tendsto_uniformly_on (complex.cderiv δ ∘ F) (complex.cderiv δ f) φ K
theorem
complex.tendsto_uniformly_on_deriv_of_cthickening_subset
{E : Type u_1}
{ι : Type u_2}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{U K : set ℂ}
{φ : filter ι}
{F : ι → ℂ → E}
{f : ℂ → E}
(hf : tendsto_locally_uniformly_on F f φ U)
(hF : ∀ᶠ (n : ι) in φ, differentiable_on ℂ (F n) U)
{δ : ℝ}
(hδ : 0 < δ)
(hK : is_compact K)
(hU : is_open U)
(hKU : metric.cthickening δ K ⊆ U) :
tendsto_uniformly_on (deriv ∘ F) (complex.cderiv δ f) φ K
theorem
complex.exists_cthickening_tendsto_uniformly_on
{E : Type u_1}
{ι : Type u_2}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{U K : set ℂ}
{φ : filter ι}
{F : ι → ℂ → E}
{f : ℂ → E}
(hf : tendsto_locally_uniformly_on F f φ U)
(hF : ∀ᶠ (n : ι) in φ, differentiable_on ℂ (F n) U)
(hK : is_compact K)
(hU : is_open U)
(hKU : K ⊆ U) :
∃ (δ : ℝ) (H : δ > 0), metric.cthickening δ K ⊆ U ∧ tendsto_uniformly_on (deriv ∘ F) (complex.cderiv δ f) φ K
theorem
tendsto_locally_uniformly_on.differentiable_on
{E : Type u_1}
{ι : Type u_2}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{U : set ℂ}
{φ : filter ι}
{F : ι → ℂ → E}
{f : ℂ → E}
[φ.ne_bot]
(hf : tendsto_locally_uniformly_on F f φ U)
(hF : ∀ᶠ (n : ι) in φ, differentiable_on ℂ (F n) U)
(hU : is_open U) :
differentiable_on 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 tendsto_locally_uniformly_on.deriv).
theorem
tendsto_locally_uniformly_on.deriv
{E : Type u_1}
{ι : Type u_2}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{U : set ℂ}
{φ : filter ι}
{F : ι → ℂ → E}
{f : ℂ → E}
(hf : tendsto_locally_uniformly_on F f φ U)
(hF : ∀ᶠ (n : ι) in φ, differentiable_on ℂ (F n) U)
(hU : is_open U) :
tendsto_locally_uniformly_on (deriv ∘ F) (deriv f) φ U
theorem
complex.differentiable_on_tsum_of_summable_norm
{E : Type u_1}
{ι : Type u_2}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{U : set ℂ}
{F : ι → ℂ → E}
{u : ι → ℝ}
(hu : summable u)
(hf : ∀ (i : ι), differentiable_on ℂ (F i) U)
(hU : is_open U)
(hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i) :
differentiable_on ℂ (λ (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.has_sum_deriv_of_summable_norm
{E : Type u_1}
{ι : Type u_2}
[normed_add_comm_group E]
[normed_space ℂ E]
[complete_space E]
{U : set ℂ}
{z : ℂ}
{F : ι → ℂ → E}
{u : ι → ℝ}
(hu : summable u)
(hf : ∀ (i : ι), differentiable_on ℂ (F i) U)
(hU : is_open U)
(hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i)
(hz : z ∈ U) :
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.