42 lines
1.1 KiB
Plaintext
42 lines
1.1 KiB
Plaintext
import Mathlib.Analysis.Complex.CauchyIntegral
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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import Mathlib.Analysis.Analytic.IsolatedZeros
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lemma xx
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{f : ℂ → ℂ}
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{S : Set ℂ}
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(h₁S : IsPreconnected S)
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(h₂S : IsCompact S)
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(hf : ∀ s ∈ S, AnalyticAt ℂ f s) :
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∃ o : ℂ → ℕ, ∃ F : ℂ → ℂ, ∀ z ∈ S, (AnalyticAt ℂ F z) ∧ (F z ≠ 0) ∧ (f z = F z * ∏ᶠ s ∈ S, (z - s) ^ (o s)) := by
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let o : ℂ → ℕ := by
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intro z
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if hz : z ∈ S then
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let A := hf z hz
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let B := A.order
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exact (A.order : ⊤)
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else
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exact 0
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sorry
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theorem jensen_case_R_eq_one'
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(f : ℂ → ℂ)
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(h₁f : Differentiable ℂ f)
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(h₂f : f 0 ≠ 0)
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(S : Finset ℕ)
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(a : S → ℂ)
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(ha : ∀ s, a s ∈ Metric.ball 0 1)
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(F : ℂ → ℂ)
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(h₁F : Differentiable ℂ F)
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(h₂F : ∀ z, F z ≠ 0)
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(h₃F : f = fun z ↦ (F z) * ∏ s : S, (z - a s))
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:
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Real.log ‖f 0‖ = -∑ s, Real.log (‖a s‖⁻¹) + (2 * Real.pi)⁻¹ * ∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖f (circleMap 0 1 x)‖ := by
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sorry
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