complexOfReal_hasDerivAt — Mathlib

English

The complexOfReal construction agrees with the original map as a function (simp lemma).

Русский

Лемма (simp): complexOfReal совпадает с исходной картой как функцией.

LaTeX

$$coe_complexOfReal(h) = id$$

Lean4

/-- In cases where the **Cauchy-Riemann Equation** guarantees complex differentiability at `x`, the
complex derivative equals `ContinuousLinearMap.complexOfReal` of the real derivative.
-/
theorem complexOfReal_hasDerivAt (h₁ : DifferentiableAt ℝ f x) (h₂ : fderiv ℝ f x I = I • fderiv ℝ f x 1) :
    HasDerivAt f ((fderiv ℝ f x).complexOfReal h₂ 1) x :=
  by
  rw [hasDerivAt_iff_hasFDerivAt, smulRight_one_one]
  exact hasFDerivAt_of_restrictScalars ℝ h₁.hasFDerivAt rfl

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