nevanlinna/Nevanlinna/holomorphic.lean

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import Mathlib.Analysis.Complex.TaylorSeries
import Nevanlinna.cauchyRiemann
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variable {E : Type*} [NormedAddCommGroup E] [NormedSpace E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace F]
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace G]
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def HolomorphicAt (f : E → F) (x : E) : Prop :=
∃ s ∈ nhds x, ∀ z ∈ s, DifferentiableAt f z
theorem HolomorphicAt_iff
{f : E → F}
{x : E} :
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HolomorphicAt f x ↔ ∃ s :
Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, DifferentiableAt f z) := by
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constructor
· intro hf
obtain ⟨t, h₁t, h₂t⟩ := hf
obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁t
use s
constructor
· assumption
· constructor
· assumption
· intro z hz
exact h₂t z (h₁s hz)
· intro hyp
obtain ⟨s, h₁s, h₂s, hf⟩ := hyp
use s
constructor
· apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
· assumption
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theorem HolomorphicAt_analyticAt
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[CompleteSpace F]
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{f : → F}
{x : } :
HolomorphicAt f x → AnalyticAt f x := by
intro hf
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
apply DifferentiableOn.analyticAt (s := s)
intro z hz
apply DifferentiableAt.differentiableWithinAt
apply h₃s
exact hz
exact IsOpen.mem_nhds h₁s h₂s
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theorem HolomorphicAt_differentiableAt
{f : E → F}
{x : E} :
HolomorphicAt f x → DifferentiableAt f x := by
intro hf
obtain ⟨s, _, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
exact h₃s x h₂s
theorem HolomorphicAt_isOpen
(f : E → F) :
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IsOpen { x : E | HolomorphicAt f x } := by
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rw [← subset_interior_iff_isOpen]
intro x hx
simp at hx
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hx
use s
constructor
· simp
constructor
· exact h₁s
· intro x hx
simp
use s
constructor
· exact IsOpen.mem_nhds h₁s hx
· exact h₃s
· exact h₂s
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theorem HolomorphicAt_comp
{g : E → F}
{f : F → G}
{z : E}
(hf : HolomorphicAt f (g z))
(hg : HolomorphicAt g z) :
HolomorphicAt (f ∘ g) z := by
obtain ⟨UE, h₁UE, h₂UE⟩ := hg
obtain ⟨UF, h₁UF, h₂UF⟩ := hf
use UE ∩ g⁻¹' UF
constructor
· simp
constructor
· assumption
· apply ContinuousAt.preimage_mem_nhds
apply (h₂UE z (mem_of_mem_nhds h₁UE)).continuousAt
assumption
· intro x hx
apply DifferentiableAt.comp
apply h₂UF
exact hx.2
apply h₂UE
exact hx.1
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theorem HolomorphicAt_neg
{f : E → F}
{z : E}
(hf : HolomorphicAt f z) :
HolomorphicAt (-f) z := by
obtain ⟨UF, h₁UF, h₂UF⟩ := hf
use UF
constructor
· assumption
· intro z hz
apply differentiableAt_neg_iff.mp
simp
exact h₂UF z hz
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theorem HolomorphicAt_contDiffAt
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[CompleteSpace F]
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{f : → F}
{z : }
(hf : HolomorphicAt f z) :
ContDiffAt 2 f z := by
let t := {x | HolomorphicAt f x}
have ht : IsOpen t := HolomorphicAt_isOpen f
have hz : z ∈ t := by exact hf
-- ContDiffAt 2 f z
apply ContDiffOn.contDiffAt _ ((IsOpen.mem_nhds_iff ht).2 hz)
suffices h : ContDiffOn 2 f t from by
apply ContDiffOn.restrict_scalars h
apply DifferentiableOn.contDiffOn _ ht
intro w hw
apply DifferentiableAt.differentiableWithinAt
exact HolomorphicAt_differentiableAt hw
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theorem CauchyRiemann'₅
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{f : → F}
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{z : }
(h : DifferentiableAt f z) :
partialDeriv Complex.I f z = Complex.I • partialDeriv 1 f z := by
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unfold partialDeriv
conv =>
left
rw [DifferentiableAt.fderiv_restrictScalars h]
simp
rw [← mul_one Complex.I]
rw [← smul_eq_mul]
conv =>
right
right
rw [DifferentiableAt.fderiv_restrictScalars h]
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simp
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theorem CauchyRiemann'₆
{f : → F}
{z : }
(h : HolomorphicAt f z) :
partialDeriv Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv 1 f := by
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obtain ⟨s, h₁s, hz, h₂f⟩ := HolomorphicAt_iff.1 h
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apply Filter.eventuallyEq_iff_exists_mem.2
use s
constructor
· exact IsOpen.mem_nhds h₁s hz
· intro w hw
apply CauchyRiemann'₅
exact h₂f w hw