| import Mathlib |
| import OAI.Analysis.Crouzeix.CircleAnalytic |
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|
| namespace OAI |
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| /-! Angular Radius. -/ |
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| noncomputable section |
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| open Set Filter Metric Topology Function Complex MeasureTheory |
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| namespace CrouzeixHilbert.Boundary |
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| lemma integral_scaled_circleCoordinate {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] (f : ℂ → E) (R : ℝ) : |
| (∫ t, f ((R : ℂ) * circleCoordinate t) ∂circleMeasure) = Real.circleAverage f 0 R := by |
| rw [integral_circleCoordinate_eq_circleAverage (fun z => f ((R : ℂ) * z))] |
| unfold Real.circleAverage |
| congr 1 |
| apply intervalIntegral.integral_congr |
| intro t ht |
| simp only [circleMap, zero_add, ofReal_one, one_mul] |
|
|
| lemma integral_scaled_coordinate_cauchy {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℂ E] (f : ℂ → E) {R : ℝ} (hR : 0 < R) : |
| (∫ t, ((R : ℂ) * circleCoordinate t) • f ((R : ℂ) * circleCoordinate t) |
| ∂circleMeasure) = |
| (2 * (Real.pi : ℂ) * I)⁻¹ • (∮ z in C(0, R), f z) := by |
| rw [integral_scaled_circleCoordinate (fun z => z • f z) R, |
| Real.circleAverage_eq_circleIntegral hR.ne'] |
| congr 1 |
| apply circleIntegral.integral_congr hR.le |
| intro z hz |
| have hn : z ≠ 0 := norm_pos_iff.mp (by rw [mem_sphere_zero_iff_norm.mp hz]; exact hR) |
| simp only [sub_zero, smul_smul, inv_mul_cancel₀ hn, one_smul] |
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| lemma integral_scaled_deform {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℂ E] {r R : ℝ} (hr : 0 < r) (hrR : r ≤ R) {f : ℂ → E} |
| (hc : ContinuousOn f (closedBall 0 R \ ball 0 r)) |
| (hd : DifferentiableOn ℂ f (ball 0 R \ closedBall 0 r)) : |
| (∫ t, ((R : ℂ) * circleCoordinate t) • f ((R : ℂ) * circleCoordinate t) |
| ∂circleMeasure) = |
| ∫ t, ((r : ℂ) * circleCoordinate t) • f ((r : ℂ) * circleCoordinate t) |
| ∂circleMeasure := by |
| rw [integral_scaled_coordinate_cauchy f (hr.trans_le hrR), |
| integral_scaled_coordinate_cauchy f hr] |
| congr 1 |
| apply Complex.circleIntegral_eq_of_differentiable_on_annulus_off_countable hr hrR countable_empty hc |
| intro z hz |
| exact (hd z hz.1).differentiableAt ((isOpen_ball.sdiff isClosed_closedBall).mem_nhds hz.1) |
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| lemma integral_laurent_deform {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] |
| {r R : ℝ} (hr : 0 < r) (hrR : r ≤ R) {f : ℂ → E} |
| (hc : ContinuousOn f (closedBall 0 R \ ball 0 r)) |
| (hd : DifferentiableOn ℂ f (ball 0 R \ closedBall 0 r)) (j : ℤ) : |
| (∫ t, (((R : ℂ) * circleCoordinate t)^j) • f ((R : ℂ) * circleCoordinate t) |
| ∂circleMeasure) = |
| ∫ t, (((r : ℂ) * circleCoordinate t)^j) • f ((r : ℂ) * circleCoordinate t) |
| ∂circleMeasure := by |
| have hn {z : ℂ} (hz : z ∈ closedBall 0 R \ ball 0 r) : z ≠ 0 := by |
| have hle : r ≤ ‖z‖ := not_lt.mp (mt mem_ball_zero_iff.mpr hz.2) |
| exact norm_pos_iff.mp (hr.trans_le hle) |
| have hh := integral_scaled_deform hr hrR |
| (f := fun z => z^(j-1) • f z) |
| ((continuousOn_id.zpow₀ (j-1) (fun z hz => Or.inl (hn hz))).smul hc) |
| ((((differentiableOn_id (𝕜 := ℂ)).zpow (Or.inl (fun z hz => hn ⟨ball_subset_closedBall hz.1, |
| fun h => hz.2 (ball_subset_closedBall h)⟩)))).smul hd) |
| have hs {z : ℂ} (hz : z ≠ 0) : z * z^(j-1) = z^j := by |
| rw [← zpow_one_add₀ hz] |
| congr 1 |
| omega |
| have hRz (t : CircleSpace) : (R : ℂ) * circleCoordinate t ≠ 0 := |
| mul_ne_zero (ofReal_ne_zero.mpr (hr.trans_le hrR).ne') (circleCoordinate_ne_zero t) |
| have hrz (t : CircleSpace) : (r : ℂ) * circleCoordinate t ≠ 0 := |
| mul_ne_zero (ofReal_ne_zero.mpr hr.ne') (circleCoordinate_ne_zero t) |
| simpa only [smul_smul, hs (hRz _), hs (hrz _)] using hh |
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| end CrouzeixHilbert.Boundary |
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| end |
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| end OAI |
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