This commit is contained in:
Stefan Kebekus 2024-08-05 14:18:02 +02:00
parent 854b7ef492
commit 4387149e33
1 changed files with 33 additions and 42 deletions

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@ -289,47 +289,6 @@ theorem primitive_hasDerivAtBasepoint
exact hasDerivAt_id z₀ exact hasDerivAt_id z₀
lemma integrability₁
{E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E]
(f : → E)
(hf : Differentiable f)
(a₁ a₂ b : ) :
IntervalIntegrable (fun x => f { re := x, im := b }) MeasureTheory.volume a₁ a₂ := by
apply Continuous.intervalIntegrable
apply Continuous.comp
exact Differentiable.continuous hf
have : ((fun x => { re := x, im := b }) : ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
funext x
apply Complex.ext
rw [Complex.add_re]
simp
rw [Complex.add_im]
simp
rw [this]
continuity
lemma integrability₂
{E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E]
(f : → E)
(hf : Differentiable f)
(a₁ a₂ b : ) :
IntervalIntegrable (fun x => f { re := b, im := x }) MeasureTheory.volume a₁ a₂ := by
apply Continuous.intervalIntegrable
apply Continuous.comp
exact Differentiable.continuous hf
have : (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
funext x
apply Complex.ext
rw [Complex.add_re]
simp
simp
rw [this]
apply Continuous.add
continuity
fun_prop
theorem primitive_additivity theorem primitive_additivity
{E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E]
(f : → E) (f : → E)
@ -545,8 +504,40 @@ theorem primitive_additivity'
(hz₁ : z₁ ∈ Metric.ball z₀ R) (hz₁ : z₁ ∈ Metric.ball z₀ R)
: :
primitive z₀ f =ᶠ[nhds z₁] fun z ↦ (primitive z₁ f z) + (primitive z₀ f z₁) := by primitive z₀ f =ᶠ[nhds z₁] fun z ↦ (primitive z₁ f z) + (primitive z₀ f z₁) := by
let ε := R - dist z₀ z₁
let rx := dist z₀.re z₁.re + ε/(2 : )
let ry := dist z₀.im z₁.im + ε/(2 : )
have h'ry : 0 < ry := by
sorry sorry
have h'f : DifferentiableOn f (Metric.ball z₀.re rx × Metric.ball z₀.im ry) := by
sorry
have h'z₁ : z₁ ∈ (Metric.ball z₀.re rx × Metric.ball z₀.im ry) := by
sorry
let A := primitive_additivity f z₀ rx ry h'ry h'f z₁ h'z₁
obtain ⟨εx, εy, hε⟩ := A
apply Filter.eventuallyEq_iff_exists_mem.2
use (Metric.ball z₁.re εx × Metric.ball z₁.im εy)
constructor
· apply IsOpen.mem_nhds
apply IsOpen.reProdIm
exact Metric.isOpen_ball
exact Metric.isOpen_ball
constructor
· simp
sorry
· simp
sorry
· intro x hx
simp
rw [← sub_zero (primitive z₀ f x), ← hε x hx]
abel
theorem primitive_hasDerivAt theorem primitive_hasDerivAt
{E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] {E : Type u} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E]