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refactor: golf Gaussian/GaussianIntegral #38250

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refactor: golf Gaussian/GaussianIntegral #38250

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refactor: golf Gaussian/GaussianIntegral
yuanyi-350 Apr 19, 2026
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golf
yuanyi-350 Apr 19, 2026
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29 changes: 9 additions & 20 deletions Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
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Expand Up @@ -288,27 +288,16 @@ theorem integral_gaussian_complex {b : ℂ} (hb : 0 < re b) :
-- The Gaussian integral on the half-line, `∫ x in Ioi 0, exp (-b * x^2)`, for complex `b`.
theorem integral_gaussian_complex_Ioi {b : ℂ} (hb : 0 < re b) :
∫ x : ℝ in Ioi 0, cexp (-b * (x : ℂ) ^ 2) = (π / b) ^ (1 / 2 : ℂ) / 2 := by
let f : ℝ → ℂ := fun x => cexp (-b * (x : ℂ) ^ 2)
have full_integral := integral_gaussian_complex hb
have : MeasurableSet (Ioi (0 : ℝ)) := measurableSet_Ioi
rw [← integral_add_compl this (integrable_cexp_neg_mul_sq hb), compl_Ioi] at full_integral
suffices ∫ x : ℝ in Iic 0, cexp (-b * (x : ℂ) ^ 2) = ∫ x : ℝ in Ioi 0, cexp (-b * (x : ℂ) ^ 2) by
rw [this, ← mul_two] at full_integral
rwa [eq_div_iff]; exact two_ne_zero
have : ∀ c : ℝ, ∫ x in (0 : ℝ)..c, cexp (-b * (x : ℂ) ^ 2) =
∫ x in -c..0, cexp (-b * (x : ℂ) ^ 2) := by
intro c
have := intervalIntegral.integral_comp_sub_left (a := 0) (b := c)
(fun x => cexp (-b * (x : ℂ) ^ 2)) 0
simpa [zero_sub, neg_sq, neg_zero] using this
have t1 :=
intervalIntegral_tendsto_integral_Ioi 0 (integrable_cexp_neg_mul_sq hb).integrableOn tendsto_id
have t2 :
Tendsto (fun c : ℝ => ∫ x : ℝ in (0 : ℝ)..c, cexp (-b * (x : ℂ) ^ 2)) atTop
(𝓝 (∫ x : ℝ in Iic 0, cexp (-b * (x : ℂ) ^ 2))) := by
simp_rw [this]
refine intervalIntegral_tendsto_integral_Iic _ ?_ tendsto_neg_atTop_atBot
apply (integrable_cexp_neg_mul_sq hb).integrableOn
exact tendsto_nhds_unique t2 t1
have h_eq : ∫ x : ℝ in Iic 0, f x = ∫ x : ℝ in Ioi 0, f x := by
calc
_ = ∫ x : ℝ in Ioi 0, f (-x) := by simp
_ = _ := setIntegral_congr_fun measurableSet_Ioi fun x hx ↦ by simp [f]
have hmeas : MeasurableSet (Ioi (0 : ℝ)) := measurableSet_Ioi
rw [← integral_add_compl hmeas (integrable_cexp_neg_mul_sq hb), compl_Ioi, h_eq, ← mul_two]
at full_integral
exact (eq_div_iff two_ne_zero).2 (by simpa using full_integral)

-- The Gaussian integral on the half-line, `∫ x in Ioi 0, exp (-b * x^2)`, for real `b`.
theorem integral_gaussian_Ioi (b : ℝ) :
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