hasSum_taylorSeries_of_entire — Mathlib

English

If f maps ball B(0,R) to itself and f(0)=0, then ‖f(z)‖ ≤ ‖z‖ for z in B(0,R).

Русский

Если f отображает шар B(0,R) в сам шар и f(0)=0, то ‖f(z)‖ ≤ ‖z‖ для всех z в шаре.

LaTeX

$$$\|f(z)\| \le \|z\|$$$

Lean4

/-- A function that is complex differentiable on the complex plane is given by evaluating
its Taylor series at any point `c`. -/
theorem hasSum_taylorSeries_of_entire : HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) :=
  hasSum_taylorSeries_on_emetric_ball hf.differentiableOn <| EMetric.mem_ball.mpr <| edist_lt_top ..

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