Theorems · Theorem · harmonic analysis
integral_cexp_quadratic
∀ {b : ℂ},
b.re < 0 →
∀ (c d : ℂ),
∫ (x : ℝ), Complex.exp (b * ↑x ^ 2 + c * ↑x + d) = (↑Real.pi / -b) ^ (1 / 2) * Complex.exp (d - c ^ 2 / (4 * b))- Cited by
- 5 results in Mathlib
- Foundations
- Depth 292 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Complexstatement and proof · cited by 5,565
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- le_reflproof · cited by 2,061
- Nat.cast_zeroproof · cited by 1,870
- MeasureTheory.integralstatement and proof · cited by 1,779
- Real.pistatement and proof · cited by 1,774
- Complex.ofRealstatement and proof · cited by 1,654
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
- neg_negproof · cited by 960
- one_ne_zeroproof · cited by 885
Cited by5
Results whose statement or proof uses this declaration.
- ProbabilityTheory.complexMGF_id_gaussianRealproof · cited by 3
- fourier_gaussian_pi'proof · cited by 2
- integrable_cexp_quadratic'proof · cited by 1
- GaussianFourier.integral_cexp_neg_sum_mul_addproof · cited by 1
- fourierIntegral_gaussianproof · cited by 0