diff --git a/Nevanlinna/holomorphic.examples.lean b/Nevanlinna/holomorphic_examples.lean similarity index 93% rename from Nevanlinna/holomorphic.examples.lean rename to Nevanlinna/holomorphic_examples.lean index 26972ee..f4062d2 100644 --- a/Nevanlinna/holomorphic.examples.lean +++ b/Nevanlinna/holomorphic_examples.lean @@ -1,7 +1,7 @@ import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv import Nevanlinna.complexHarmonic -import Nevanlinna.complexHarmonic import Nevanlinna.holomorphicAt +import Nevanlinna.holomorphic_primitive theorem CauchyRiemann₆ @@ -194,7 +194,20 @@ theorem harmonic_is_realOfHolomorphic apply Differentiable.const_smul exact reg₁f_I.differentiable le_rfl - - - - sorry + let F := primitive 0 g + use F + intro z + constructor + · -- HolomorphicAt F z + apply HolomorphicAt_iff.2 + use {z : ℂ | true} + constructor + · exact isOpen_const + · constructor + · simp + · intro w hw + let A : HasDerivAt (primitive 0 g) (g w) w := primitive_fderiv g reg₁ + apply A.differentiableAt + · -- (F z).re = f z + + sorry diff --git a/Nevanlinna/holomorphic.primitive.lean b/Nevanlinna/holomorphic_primitive.lean similarity index 98% rename from Nevanlinna/holomorphic.primitive.lean rename to Nevanlinna/holomorphic_primitive.lean index c2a67be..4e7a4de 100644 --- a/Nevanlinna/holomorphic.primitive.lean +++ b/Nevanlinna/holomorphic_primitive.lean @@ -3,8 +3,10 @@ import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.MeasureTheory.Integral.DivergenceTheorem import Mathlib.MeasureTheory.Integral.IntervalIntegral import Mathlib.MeasureTheory.Function.LocallyIntegrable +import Nevanlinna.partialDeriv +import Nevanlinna.cauchyRiemann - +/- noncomputable def partialDeriv {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] : E → (E → F) → (E → F) := @@ -47,7 +49,6 @@ theorem partialDeriv_compCLE rw [ContinuousLinearEquiv.comp_differentiableAt_iff] exact hyp - theorem partialDeriv_smul'₂ {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] @@ -84,11 +85,10 @@ theorem partialDeriv_smul'₂ rw [partialDeriv_compCLE] tauto - theorem CauchyRiemann₄ {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : ℂ → F} : - (Differentiable ℂ f) → partialDeriv Complex.I f = Complex.I • partialDeriv 1 f := by + (Differentiable ℂ f) → partialDeriv ℝ Complex.I f = Complex.I • partialDeriv ℝ 1 f := by intro h unfold partialDeriv @@ -106,6 +106,7 @@ theorem CauchyRiemann₄ rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)] funext w simp +-/ theorem MeasureTheory.integral2_divergence₃ @@ -207,9 +208,9 @@ theorem integral_divergence₅ exact h₁f let A := integral_divergence₄ (-Complex.I • F) F h₁g h₁f lowerLeft.re upperRight.im upperRight.re lowerLeft.im - have {z : ℂ} : fderiv ℝ F z Complex.I = partialDeriv Complex.I F z := by rfl + have {z : ℂ} : fderiv ℝ F z Complex.I = partialDeriv ℝ Complex.I F z := by rfl conv at A in (fderiv ℝ F _) _ => rw [this] - have {z : ℂ} : fderiv ℝ (-Complex.I • F) z 1 = partialDeriv 1 (-Complex.I • F) z := by rfl + have {z : ℂ} : fderiv ℝ (-Complex.I • F) z 1 = partialDeriv ℝ 1 (-Complex.I • F) z := by rfl conv at A in (fderiv ℝ (-Complex.I • F) _) _ => rw [this] conv at A => left