56 lines
1.5 KiB
Plaintext
56 lines
1.5 KiB
Plaintext
import Mathlib.Analysis.Complex.CauchyIntegral
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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 Complex.log
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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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{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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→ ∀ z ∈ Metric.ball 0 R, f z ≠ 0
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→ (∮ (z : ℂ) in C(0, R), Complex.log ‖f z‖ ) = 2 * π * R * log ‖f 0‖ := by
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sorry
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