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This commit is contained in:
Stefan Kebekus
2024-08-01 13:51:15 +02:00
parent a1f96806a1
commit 78de1bd3b0
1 changed files with 20 additions and 12 deletions
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32
Nevanlinna/holomorphic_primitive2.lean
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@@ -1,5 +1,5 @@
import Mathlib.Analysis.Complex.TaylorSeries import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.Data.ENNReal.Basic
noncomputable def primitive noncomputable def primitive
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] : {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] :
@@ -333,23 +333,24 @@ lemma integrability₂
theorem primitive_additivity theorem primitive_additivity
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
(f : ℂ → E) (f : ℂ → E)
(hf : Differentiable ℂ f) (z₀ : ℂ)
(z₀ z₁ : ℂ) : (R : ℝ)
(fun z ↦ (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁)) = 0 := by (hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
funext z (z₁ : ℂ)
(hz₁ : z₁ ∈ Metric.ball z₀ R) :
∀ z ∈ Metric.ball z₁ (R - ‖z₁‖), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
intro z _
unfold primitive unfold primitive
have : (∫ (x : ℝ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ℝ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ℝ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by have : (∫ (x : ℝ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ℝ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ℝ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by
rw [intervalIntegral.integral_add_adjacent_intervals] rw [intervalIntegral.integral_add_adjacent_intervals]
apply integrability₁ f hf sorry --apply integrability₁ f hf
apply integrability₁ f hf sorry --apply integrability₁ f hf
rw [this] rw [this]
have : (∫ (x : ℝ) in z₀.im..z.im, f { re := z.re, im := x }) = (∫ (x : ℝ) in z₀.im..z₁.im, f { re := z.re, im := x }) + (∫ (x : ℝ) in z₁.im..z.im, f { re := z.re, im := x }) := by have : (∫ (x : ℝ) in z₀.im..z.im, f { re := z.re, im := x }) = (∫ (x : ℝ) in z₀.im..z₁.im, f { re := z.re, im := x }) + (∫ (x : ℝ) in z₁.im..z.im, f { re := z.re, im := x }) := by
rw [intervalIntegral.integral_add_adjacent_intervals] rw [intervalIntegral.integral_add_adjacent_intervals]
apply integrability₂ f hf sorry --apply integrability₂ f hf
apply integrability₂ f hf sorry --apply integrability₂ f hf
rw [this] rw [this]
simp simp
@@ -358,7 +359,14 @@ theorem primitive_additivity
rw [this] rw [this]
simp simp
let A := Complex.integral_boundary_rect_eq_zero_of_differentiableOn f ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩ (hf.differentiableOn) have H : DifferentiableOn ℂ f (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) := by
apply DifferentiableOn.mono hf
intro x hx
simp
sorry
let A := Complex.integral_boundary_rect_eq_zero_of_differentiableOn f ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩ H
have {x : ℝ} {w : ℂ} : ↑x + w.im * Complex.I = { re := x, im := w.im } := by have {x : ℝ} {w : ℂ} : ↑x + w.im * Complex.I = { re := x, im := w.im } := by
apply Complex.ext apply Complex.ext
· simp · simp