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ef96466 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralRegularPhysical
namespace OAI
/-! Holomorphy is preserved when restoring the angular radial factor. -/
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem spectralAngularPair_analyticAt (ell : ℕ) (Y : ℂ → ℝ → E₄)
(z : ℂ) (r : ℝ) (hY : AnalyticAt ℂ (fun lam => Y lam r) z) :
AnalyticAt ℂ (fun lam => spectralAngularPair ell (Y lam) r) z := by
have h1 : AnalyticAt ℂ (fun lam => (Y lam r).1) z := by
convert! ((ContinuousLinearMap.fst ℂ (ℂ × ℂ) (ℂ × ℂ)).analyticAt _).comp hY
have h2 : AnalyticAt ℂ (fun lam => (Y lam r).2) z := by
convert! ((ContinuousLinearMap.snd ℂ (ℂ × ℂ) (ℂ × ℂ)).analyticAt _).comp hY
have h11 : AnalyticAt ℂ (fun lam => (Y lam r).1.1) z := by
convert! ((ContinuousLinearMap.fst ℂ ℂ ℂ).analyticAt _).comp h1
have h12 : AnalyticAt ℂ (fun lam => (Y lam r).1.2) z := by
convert! ((ContinuousLinearMap.snd ℂ ℂ ℂ).analyticAt _).comp h1
have h21 : AnalyticAt ℂ (fun lam => (Y lam r).2.1) z := by
convert! ((ContinuousLinearMap.fst ℂ ℂ ℂ).analyticAt _).comp h2
have h22 : AnalyticAt ℂ (fun lam => (Y lam r).2.2) z := by
convert! ((ContinuousLinearMap.snd ℂ ℂ ℂ).analyticAt _).comp h2
exact ((analyticAt_const.mul h11).prod
(analyticAt_const.mul ((analyticAt_const.mul h11).add h12))).prod
((analyticAt_const.mul h21).prod
(analyticAt_const.mul ((analyticAt_const.mul h21).add h22)))
end DefocusingNLS
end OAI
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