Working…

This commit is contained in:
Stefan Kebekus
2024-04-23 10:56:52 +02:00
parent e37ef6da16
commit f33953de0a

View File

@@ -2,6 +2,7 @@ import Mathlib.Analysis.Complex.CauchyIntegral
#check DiffContOnCl.circleIntegral_sub_inv_smul
open Real
theorem CauchyIntegralFormula :
@@ -10,8 +11,45 @@ theorem CauchyIntegralFormula :
{f : }, -- Holomorphic function
DiffContOnCl f (Metric.ball 0 R)
w Metric.ball 0 R
( (z : ) in C(0, R), (z - w)⁻¹ f z) = (2 * Real.pi * Complex.I) f w := by
( (z : ) in C(0, R), (z - w)⁻¹ f z) = (2 * π * Complex.I) f w := by
exact DiffContOnCl.circleIntegral_sub_inv_smul
#check CauchyIntegralFormula
#check HasDerivAt.continuousAt
#check Real.log
#check Complex.log
#check Complex.exp
theorem SimpleCauchyFormula :
{R : } -- Radius of the ball
{w : } -- Point in the interior of the ball
{f : }, -- Holomorphic function
Differentiable f
w Metric.ball 0 R
( (z : ) in C(0, R), (z - w)⁻¹ f z) = (2 * Real.pi * Complex.I) f w := by
intro R w f fHyp
apply DiffContOnCl.circleIntegral_sub_inv_smul
constructor
· exact Differentiable.differentiableOn fHyp
· suffices Continuous f from by
exact Continuous.continuousOn this
rw [continuous_iff_continuousAt]
intro x
exact DifferentiableAt.continuousAt (fHyp x)
theorem JensenFormula₂ :
{R : } -- Radius of the ball
{w : } -- Point in the interior of the ball
{f : }, -- Holomorphic function
Differentiable f
z Metric.ball 0 R, f z 0
( (z : ) in C(0, R), Complex.log f z ) = 2 * π * R * log f 0 := by
sorry