diff --git a/nevanlinna/test.lean b/nevanlinna/test.lean index 0cd2534..1bf0961 100644 --- a/nevanlinna/test.lean +++ b/nevanlinna/test.lean @@ -46,10 +46,33 @@ theorem SimpleCauchyFormula : theorem JensenFormula₂ : ∀ {R : ℝ} -- Radius of the ball - {w : ℂ} -- Point in the interior of the ball {f : ℂ → ℂ}, -- Holomorphic function Differentiable ℂ f - → ∀ z ∈ Metric.ball 0 R, f z ≠ 0 - → (∮ (z : ℂ) in C(0, R), Complex.log ‖f z‖ ) = 2 * π * R * log ‖f 0‖ := by + → (∀ z ∈ Metric.ball 0 R, f z ≠ 0) + → (∫ (θ : ℝ) in Set.Icc 0 (2 * π), Complex.log ‖f (circleMap 0 R θ)‖ ) = 2 * π * log ‖f 0‖ := by + + intro r f fHyp₁ fHyp₂ + + have : (fun θ ↦ Complex.log ↑‖f (circleMap 0 r θ)‖) = (fun θ ↦ ((deriv (circleMap 0 r) θ)) • ((deriv (circleMap 0 r) θ)⁻¹ • Complex.log ↑‖f (circleMap 0 r θ)‖)) := by + funext θ + rw [← smul_assoc] + + rw [smul_eq_mul, smul_eq_mul] + rw [mul_inv_cancel, one_mul] + + simp + + sorry + + rw [this] + + simp + + let XX := circleIntegral_def_Icc (fun z ↦ -(Complex.I * z⁻¹ * (Complex.log ↑‖f z‖))) 0 r + simp at XX + + rw [← XX] + + sorry