differentiableAt_iteratedDerivWithin_cexp

English

The function r1 assigns to a point z in the upper half-plane the value z.im^2 / (z.re^2 + z.im^2).

Русский

Функция r1 м аппрет по z на верхнюю половину плоскости как z.im^2 / (z.re^2 + z.im^2).

LaTeX

$$$r1(z) = \dfrac{(\operatorname{Im} z)^2}{(\operatorname{Re} z)^2 + (\operatorname{Im} z)^2}.$$$

Lean4

theorem differentiableAt_iteratedDerivWithin_cexp (n a : ℕ) {s : Set ℂ} (hs : IsOpen s) {r : ℂ} (hr : r ∈ s) :
    DifferentiableAt ℂ (iteratedDerivWithin a (fun z ↦ cexp (2 * π * I * z) ^ n) s) r :=
  by
  apply DifferentiableOn.differentiableAt _ (hs.mem_nhds hr)
  suffices DifferentiableOn ℂ (iteratedDeriv a (fun z ↦ cexp (2 * π * I * z) ^ n)) s by
    apply this.congr (iteratedDerivWithin_of_isOpen hs)
  fun_prop

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