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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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAngularMeasure
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralRadialWeightedIntegral
namespace OAI
/-! The angular term uses r^9 dr while the radial terms use r^11 dr. -/
open Set MeasureTheory
namespace DefocusingNLS
theorem spectral_angular_complex_integral (R : ℝ) (hR : 0 ≤ R) (f : ℝ → ℂ) :
(∫ r, f r ∂spectralAngularMeasure R)=∫ r in (0 : ℝ)..R, (r : ℂ)^9*f r := by
unfold spectralAngularMeasure
rw [integral_withDensity_eq_integral_toReal_smul (by fun_prop)
(Filter.Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top)),
intervalIntegral.integral_of_le hR]
calc
_ = ∫ r in Icc (0 : ℝ) R, (r : ℂ)^9*f r := by
apply setIntegral_congr_fun measurableSet_Icc
intro r hr
change (ENNReal.ofReal ((max r 0)^9)).toReal • f r=(r : ℂ)^9*f r
rw [ENNReal.toReal_ofReal (by positivity),max_eq_left hr.1]
simp only [Complex.real_smul,Complex.ofReal_pow]
_ = _ := integral_Icc_eq_integral_Ioc
end DefocusingNLS
end OAI
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