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This commit is contained in:
Stefan Kebekus 2024-07-30 15:11:57 +02:00
parent 1d27eeb66b
commit 8d100b2333
1 changed files with 30 additions and 11 deletions
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41
Nevanlinna/holomorphic_examples.lean
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@ -1,4 +1,3 @@
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Nevanlinna.complexHarmonic import Nevanlinna.complexHarmonic
import Nevanlinna.holomorphicAt import Nevanlinna.holomorphicAt
import Nevanlinna.holomorphic_primitive import Nevanlinna.holomorphic_primitive
@ -194,33 +193,53 @@ theorem harmonic_is_realOfHolomorphic
apply Differentiable.const_smul apply Differentiable.const_smul
exact reg₁f_I.differentiable le_rfl exact reg₁f_I.differentiable le_rfl
let F := primitive 0 g let F := fun z ↦ (primitive 0 g) z + f 0
have regF : Differentiable ℂ F := by
apply Differentiable.add
intro x
let A : HasDerivAt (primitive 0 g) (g x) x := primitive_fderiv g reg₁
exact A.differentiableAt
simp
have pF : ∀ x a, (fderiv ℝ F x) a = (g x) * a := by have pF' : ∀ x, (fderiv ℂ F x) = ContinuousLinearMap.lsmul ℂ ℂ (g x) := by
sorry intro x
dsimp [F]
rw [fderiv_add_const]
let A : HasDerivAt (primitive 0 g) (g x) x := primitive_fderiv g reg₁
let B : HasFDerivAt (primitive 0 g) (ContinuousLinearMap.lsmul ℂ ℂ (g x)) x := by
rw [hasFDerivAt_iff_hasDerivAt]
simp
exact A
exact HasFDerivAt.fderiv B
have regF : Differentiable ℂ F := by sorry have pF'' : ∀ x, (fderiv ℝ F x) = ContinuousLinearMap.lsmul ℝ ℂ (g x) := by
intro x
rw [DifferentiableAt.fderiv_restrictScalars ℝ (regF x), pF' x]
exact rfl
use F use F
intro z intro z
constructor constructor
· -- HolomorphicAt F z · -- HolomorphicAt F z
apply HolomorphicAt_iff.2 apply HolomorphicAt_iff.2
use {z : ℂ | true} use Set.univ
constructor constructor
· exact isOpen_const · exact isOpen_const
· constructor · constructor
· simp · simp
· intro w hw · intro w _
let A : HasDerivAt (primitive 0 g) (g w) w := primitive_fderiv g reg₁ exact regF w
apply A.differentiableAt
· -- (F z).re = f z · -- (F z).re = f z
have A := reg₂f.differentiable one_le_two have A := reg₂f.differentiable one_le_two
have B : Differentiable ℝ (Complex.reCLM ∘ F) := by have B : Differentiable ℝ (Complex.reCLM ∘ F) := by
apply Differentiable.comp apply Differentiable.comp
exact ContinuousLinearMap.differentiable Complex.reCLM exact ContinuousLinearMap.differentiable Complex.reCLM
exact Differentiable.restrictScalars ℝ regF exact Differentiable.restrictScalars ℝ regF
have C : (F 0).re = f 0 := by sorry have C : (F 0).re = f 0 := by
dsimp [F]
rw [primitive_zeroAtBasepoint]
simp
apply eq_of_fderiv_eq B A _ 0 C apply eq_of_fderiv_eq B A _ 0 C
intro x intro x
rw [fderiv.comp] rw [fderiv.comp]
@ -228,7 +247,7 @@ theorem harmonic_is_realOfHolomorphic
apply ContinuousLinearMap.ext apply ContinuousLinearMap.ext
intro w intro w
simp simp
rw [pF] rw [pF'']
dsimp [g, f_1, f_I, partialDeriv] dsimp [g, f_1, f_I, partialDeriv]
simp simp
have : w = w.re • 1 + w.im • Complex.I := by simp have : w = w.re • 1 + w.im • Complex.I := by simp