diff --git a/Nevanlinna/holomorphic.primitive.lean b/Nevanlinna/holomorphic.primitive.lean index 07b8d1d..4813e2c 100644 --- a/Nevanlinna/holomorphic.primitive.lean +++ b/Nevanlinna/holomorphic.primitive.lean @@ -161,15 +161,21 @@ theorem integral_divergence₅ (∫ (x : ℝ) in a₁..b₁, Complex.I • F { re := x, im := a₂ }) + (∫ (y : ℝ) in a₂..b₂, F { re := a₁, im := y }) := by - let f := F - let h₁f : ContDiff ℝ 1 f := by sorry + let h₁f : ContDiff ℝ 1 F := (hF.contDiff : ContDiff ℂ 1 F).restrict_scalars ℝ + let g := Complex.I • F - let h₁g : ContDiff ℝ 1 g := by sorry + let h₁g : ContDiff ℝ 1 (Complex.I • F) := by + have : Complex.I • F = fun x ↦ Complex.I • F x := by rfl + rw [this] + apply ContDiff.comp + exact contDiff_const_smul Complex.I + exact h₁f - let A := integral_divergence₄ f g h₁f h₁g a₁ a₂ b₁ b₂ - have {z : ℂ} : fderiv ℝ f z 1 = partialDeriv ℝ 1 f z := by rfl - conv at A in (fderiv ℝ f _) 1 => rw [this] + let A := integral_divergence₄ F g h₁f h₁g a₁ a₂ b₁ b₂ + + have {z : ℂ} : fderiv ℝ F z 1 = partialDeriv ℝ 1 F z := by rfl + conv at A in (fderiv ℝ F _) 1 => rw [this] have {z : ℂ} : fderiv ℝ g z Complex.I = partialDeriv ℝ Complex.I g z := by rfl conv at A in (fderiv ℝ g _) Complex.I => rw [this] @@ -186,4 +192,6 @@ theorem integral_divergence₅ simp simp at A + + sorry