131 lines
3.4 KiB
Plaintext
131 lines
3.4 KiB
Plaintext
import Mathlib.Analysis.Complex.CauchyIntegral
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--import Mathlib.Analysis.Complex.Module
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open ComplexConjugate
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#check DiffContOnCl.circleIntegral_sub_inv_smul
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open Real
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theorem CauchyIntegralFormula :
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∀
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{R : ℝ} -- Radius of the ball
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{w : ℂ} -- Point in the interior of the ball
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{f : ℂ → ℂ}, -- Holomorphic function
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DiffContOnCl ℂ f (Metric.ball 0 R)
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→ w ∈ Metric.ball 0 R
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→ (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * π * Complex.I) • f w := by
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exact DiffContOnCl.circleIntegral_sub_inv_smul
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#check CauchyIntegralFormula
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#check HasDerivAt.continuousAt
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#check Real.log
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#check ComplexConjugate.conj
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#check Complex.exp
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theorem SimpleCauchyFormula :
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∀
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{R : ℝ} -- Radius of the ball
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{w : ℂ} -- Point in the interior of the ball
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{f : ℂ → ℂ}, -- Holomorphic function
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Differentiable ℂ f
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→ w ∈ Metric.ball 0 R
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→ (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * Real.pi * Complex.I) • f w := by
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intro R w f fHyp
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apply DiffContOnCl.circleIntegral_sub_inv_smul
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constructor
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· exact Differentiable.differentiableOn fHyp
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· suffices Continuous f from by
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exact Continuous.continuousOn this
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rw [continuous_iff_continuousAt]
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intro x
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exact DifferentiableAt.continuousAt (fHyp x)
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theorem JensenFormula₂ :
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∀
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{R : ℝ} -- Radius of the ball
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{f : ℂ → ℂ}, -- Holomorphic function
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Differentiable ℂ f
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→ (∀ z ∈ Metric.ball 0 R, f z ≠ 0)
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→ (∫ (θ : ℝ) in Set.Icc 0 (2 * π), Complex.log ‖f (circleMap 0 R θ)‖ ) = 2 * π * Complex.log ‖f 0‖ := by
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intro r f fHyp₁ fHyp₂
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/- We treat the trivial case r = 0 separately. -/
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by_cases rHyp : r = 0
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rw [rHyp]
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simp
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left
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unfold ENNReal.ofReal
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simp
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rw [max_eq_left (mul_nonneg zero_le_two pi_nonneg)]
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simp
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/- From hereon, we assume that r ≠ 0. -/
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/- Replace the integral over 0 … 2π by a circle integral -/
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suffices (∮ (z : ℂ) in C(0, r), -(Complex.I * z⁻¹ * Complex.log ↑(Complex.abs (f z)))) = 2 * ↑π * Complex.log ↑‖f 0‖ from by
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have : ∫ (θ : ℝ) in Set.Icc 0 (2 * π), Complex.log ↑‖f (circleMap 0 r θ)‖ = (∮ (z : ℂ) in C(0, r), -(Complex.I * z⁻¹ * Complex.log ↑(Complex.abs (f z)))) := by
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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
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funext θ
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rw [← smul_assoc, smul_eq_mul, smul_eq_mul, mul_inv_cancel, one_mul]
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simp
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exact rHyp
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rw [this]
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simp
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let tmp := circleIntegral_def_Icc (fun z ↦ -(Complex.I * z⁻¹ * (Complex.log ↑‖f z‖))) 0 r
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simp at tmp
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rw [← tmp]
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rw [this]
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have : ∀ z : ℂ, log (Complex.abs z) = 1/2 * Complex.log z + 1/2 * Complex.log (conj z) := by
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intro z
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by_cases argHyp : Complex.arg z = π
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· rw [Complex.log, argHyp, Complex.log]
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let ZZ := Complex.arg_conj z
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rw [argHyp] at ZZ
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simp at ZZ
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have : conj z = z := by
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apply?
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sorry
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let ZZ := Complex.log_conj z
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nth_rewrite 1 [Complex.log]
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simp
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let φ := Complex.arg z
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let r := Complex.abs z
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have : z = r * Complex.exp (φ * Complex.I) := by
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exact (Complex.abs_mul_exp_arg_mul_I z).symm
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have : Complex.log z = Complex.log r + r*Complex.I := by
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apply?
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sorry
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simp at XX
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sorry
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sorry
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