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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 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralRegularEquation
namespace OAI
/-! Restoring the angular factor in a regular radial spectral column. -/
namespace DefocusingNLS
noncomputable def spectralAngularJet (ell : ℕ) (Y : ℝ → ℂ × ℂ) (r : ℝ) : ℂ × ℂ :=
((r : ℂ)^ell*(Y r).1,
(r : ℂ)^ell*((ell : ℂ)/(r : ℂ)*(Y r).1+(Y r).2))
theorem spectralAngularPower_hasDerivAt (ell : ℕ) (r : ℝ) (hr : r ≠ 0) :
HasDerivAt (fun s : ℝ => (s : ℂ)^ell) ((r : ℂ)^ell*(ell : ℂ)/(r : ℂ)) r := by
cases ell with
| zero => simpa using hasDerivAt_const r (1 : ℂ)
| succ ell =>
convert! ((hasDerivAt_id r).ofReal_comp.pow (ell+1)) using 1
simp only [Nat.add_sub_cancel,id_eq,Complex.ofReal_one,mul_one]
rw [pow_succ]
field_simp [Complex.ofReal_ne_zero.mpr hr]
theorem spectralAngularJet_hasDerivAt (ell : ℕ) (h C F : ℂ) (Y : ℝ → ℂ × ℂ)
(r : ℝ) (hr : r ≠ 0)
(hY : HasDerivAt Y ((Y r).2,
-(((2*ell+11 : ℕ) : ℂ)/(r : ℂ)+h*Complex.I*(r : ℂ)/2)*(Y r).2-C*(Y r).1+F) r) :
HasDerivAt (spectralAngularJet ell Y)
((spectralAngularJet ell Y r).2,
-(11/(r : ℂ)+h*Complex.I*(r : ℂ)/2)*(spectralAngularJet ell Y r).2-
(C-h*Complex.I*(ell : ℂ)/2-((ell*(ell+10) : ℕ) : ℂ)/(r : ℂ)^2)*
(spectralAngularJet ell Y r).1+(r : ℂ)^ell*F) r := by
have hp := spectralAngularPower_hasDerivAt ell r hr
have hc : HasDerivAt (fun s : ℝ => (ell : ℂ)/(s : ℂ)) (-(ell : ℂ)/(r : ℂ)^2) r := by
convert! (((hasDerivAt_id r).ofReal_comp.inv (Complex.ofReal_ne_zero.mpr hr)).const_mul
(ell : ℂ)) using 1
simp only [id_eq,Complex.ofReal_one]
ring
have hv : HasDerivAt (fun s : ℝ => (s : ℂ)^ell*(Y s).1)
(spectralAngularJet ell Y r).2 r := by
apply (hp.mul hY.fst).congr_deriv
unfold spectralAngularJet
ring
have hd := (hv.prodMk (hp.mul ((hc.mul hY.fst).add hY.snd)))
apply hd.congr_deriv
apply Prod.ext
· rfl
· unfold spectralAngularJet
simp only [Pi.mul_apply,Pi.add_apply]
push_cast
field_simp [Complex.ofReal_ne_zero.mpr hr]
ring
end DefocusingNLS
end OAI
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