`BookProof.ChapterSphericalPlancherel.fourier_eq_cos_sub_sin` (f : ℝ → ℂ) (w : ℝ) : 𝓕 f w = ∫ x : ℝ, ((Real.cos (2 * π * (x * w)) : ℂ) - (Real.sin (2 * π * (x * w)) : ℂ) *...

Proved

BookProof.ChapterSphericalPlancherel.fourier_eq_cos_sub_sin

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSphericalPlancherel.

BookProof.ChapterSphericalPlancherel.fourier_eq_cos_sub_sin (f : ℝ → ℂ) (w : ℝ) : 𝓕 f w = ∫ x : ℝ, ((Real.cos (2 * π * (x * w)) : ℂ) - (Real.sin (2 * π * (x * w)) : ℂ) * Complex.I) * f x

Formalization note: Lean 4 identifier BookProof.ChapterSphericalPlancherel.fourier_eq_cos_sub_sin.

Preamble

-- Generated from ChapterSphericalPlancherel.lean — theorem BookProof.ChapterSphericalPlancherel.fourier_eq_cos_sub_sin
import Definitions.Def_ChapterSphericalBessel
import Mathlib
import Definitions.Def_ChapterSphericalPlancherel
open BookProof.ChapterSphericalPlancherel



open MeasureTheory Set Real SchwartzMap
open scoped FourierTransform ContDiff
open BookProof.ChapterSphericalBessel

Formal statement

theorem BookProof.ChapterSphericalPlancherel.fourier_eq_cos_sub_sin (f : ℝ → ℂ) (w : ℝ) :
    𝓕 f w = ∫ x : ℝ, ((Real.cos (2 * π * (x * w)) : ℂ)
      - (Real.sin (2 * π * (x * w)) : ℂ) * Complex.I) * f x := by sorry

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