File: Mathlib/Analysis/Complex/LocallyUniformLimit.lean

full_name: Complex.cderiv
code: /-- 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. -/
noncomputable def cderiv (r : ℝ) (f : ℂ → E) (z : ℂ) : E 
goal: E

full_name: Complex.cderiv_eq_deriv
code: theorem cderiv_eq_deriv (hU : IsOpen U) (hf : DifferentiableOn ℂ f U) (hr : 0 < r)
    (hzr : closedBall z r ⊆ U) : cderiv r f z = deriv f z 
goal: Complex.cderiv r f z = deriv f z

full_name: Complex.norm_cderiv_le
code: theorem norm_cderiv_le (hr : 0 < r) (hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M) :
    ‖cderiv r f z‖ ≤ M / r 
goal: ‖Complex.cderiv r f z‖ ≤ M / r

full_name: Complex.cderiv_sub
code: theorem cderiv_sub (hr : 0 < r) (hf : ContinuousOn f (sphere z r))
    (hg : ContinuousOn g (sphere z r)) : cderiv r (f - g) z = cderiv r f z - cderiv r g z 
goal: Complex.cderiv r (f - g) z = Complex.cderiv r f z - Complex.cderiv r g z

full_name: Complex.norm_cderiv_lt
code: theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
    (hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r 
goal: ‖Complex.cderiv r f z‖ < M / r

full_name: Complex.norm_cderiv_sub_lt
code: theorem norm_cderiv_sub_lt (hr : 0 < r) (hfg : ∀ w ∈ sphere z r, ‖f w - g w‖ < M)
    (hf : ContinuousOn f (sphere z r)) (hg : ContinuousOn g (sphere z r)) :
    ‖cderiv r f z - cderiv r g z‖ < M / r 
goal: ‖Complex.cderiv r f z - Complex.cderiv r g z‖ < M / r

full_name: TendstoUniformlyOn.cderiv
code: theorem _root_.TendstoUniformlyOn.cderiv (hF : TendstoUniformlyOn F f φ (cthickening δ K))
    (hδ : 0 < δ) (hFn : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K)) :
    TendstoUniformlyOn (cderiv δ ∘ F) (cderiv δ f) φ K 
goal: TendstoUniformlyOn (Complex.cderiv δ ∘ F) (Complex.cderiv δ f) φ K

full_name: Complex.tendstoUniformlyOn_deriv_of_cthickening_subset
code: theorem tendstoUniformlyOn_deriv_of_cthickening_subset (hf : TendstoLocallyUniformlyOn F f φ U)
    (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) {δ : ℝ} (hδ : 0 < δ) (hK : IsCompact K)
    (hU : IsOpen U) (hKU : cthickening δ K ⊆ U) :
    TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K 
goal: TendstoUniformlyOn (deriv ∘ F) (Complex.cderiv δ f) φ K

full_name: Complex.exists_cthickening_tendstoUniformlyOn
code: theorem exists_cthickening_tendstoUniformlyOn (hf : TendstoLocallyUniformlyOn F f φ U)
    (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
    ∃ δ > 0, cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K 
goal: ∃ δ > 0, Metric.cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (Complex.cderiv δ f) φ K

full_name: TendstoLocallyUniformlyOn.differentiableOn
code: /-- 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 _root_.TendstoLocallyUniformlyOn.differentiableOn [φ.NeBot]
    (hf : TendstoLocallyUniformlyOn F f φ U) (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U)
    (hU : IsOpen U) : DifferentiableOn ℂ f U 
goal: DifferentiableOn ℂ f U

full_name: TendstoLocallyUniformlyOn.deriv
code: theorem _root_.TendstoLocallyUniformlyOn.deriv (hf : TendstoLocallyUniformlyOn F f φ U)
    (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U) :
    TendstoLocallyUniformlyOn (deriv ∘ F) (deriv f) φ U 
goal: TendstoLocallyUniformlyOn (deriv ∘ F) (deriv f) φ U

full_name: Complex.differentiableOn_tsum_of_summable_norm
code: /-- 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 differentiableOn_tsum_of_summable_norm {u : ι → ℝ} (hu : Summable u)
    (hf : ∀ i : ι, DifferentiableOn ℂ (F i) U) (hU : IsOpen U)
    (hF_le : ∀ (i : ι) (w : ℂ), w ∈ U → ‖F i w‖ ≤ u i) :
    DifferentiableOn ℂ (fun w : ℂ => ∑' i : ι, F i w) U 
goal: DifferentiableOn ℂ (fun w => ∑' (i : ι), F i w) U

full_name: Complex.hasSum_deriv_of_summable_norm
code: /-- 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 hasSum_deriv_of_summable_norm {u : ι → ℝ} (hu : Summable u)
    (hf : ∀ i : ι, DifferentiableOn ℂ (F i) U) (hU : IsOpen U)
    (hF_le : ∀ (i : ι) (w : ℂ), 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) 
goal: HasSum (fun i => deriv (F i) z) (deriv (fun w => ∑' (i : ι), F i w) z)

full_name: Complex.logDeriv_tendsto
code: /-- The logarithmic derivative of a sequence of functions converging locally uniformly to a
function is the logarithmic derivative of the limit function. -/
theorem logDeriv_tendsto {ι : Type*} {p : Filter ι} (f : ι → ℂ → ℂ) (g : ℂ → ℂ)
    {s : Set ℂ} (hs : IsOpen s) (x : s) (hF : TendstoLocallyUniformlyOn f g p s)
    (hf : ∀ᶠ n : ι in p, DifferentiableOn ℂ (f n) s) (hg : g x ≠ 0) :
    Tendsto (fun n : ι => logDeriv (f n) x) p (𝓝 ((logDeriv g) x)) 
goal: Filter.Tendsto (fun n => logDeriv (f n) ↑x) p (𝓝 (logDeriv g ↑x))