circleAverage_neg_radius — Mathlib

English

If R ≠ 0, circleAverage equals a complex-contour integral expression with the factor (2πi)^{-1}.

Русский

Если R ≠ 0, circleAverage выражается через комплексный контурный интеграл с множителем (2πi)^{-1}.

LaTeX

$$$circleAverage\ f\ c\ R = (2\pi i)^{-1} \oint_{|z-c|=R} (z-c)^{-1} f(z)\,dz$$$

Lean4

/-- Circle averages do not change when replacing the radius by its negative. -/
@[simp]
theorem circleAverage_neg_radius : circleAverage f c (-R) = circleAverage f c R :=
  by
  unfold circleAverage
  simp_rw [circleMap_neg_radius, ← circleAverage_def, circleAverage_eq_integral_add π]

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