Theorems · Theorem · special functions
Complex.GammaIntegral_convergent
∀ {s : ℂ},
0 < s.re → MeasureTheory.IntegrableOn (fun x => ↑(Real.exp (-x)) * ↑x ^ (s - 1)) (Set.Ioi 0) MeasureTheory.volumeThe integral defining the Γ function converges for complex s with 0 < re s.
This is proved by reduction to the real case.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 272 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites33
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
- Norm.normproof · cited by 5,413
- Filter.univ_mem'proof · cited by 1,672
- Complex.ofRealstatement and proof · cited by 1,654
- Set.Ioistatement and proof · cited by 1,463
- MeasureTheory.MeasureSpace.volumestatement · cited by 1,323
- le_of_ltproof · cited by 1,175
- Complex.restatement and proof · cited by 882
- Real.expstatement and proof · cited by 871
- ContinuousAtproof · cited by 697
- MeasureTheory.IntegrableOnstatement · cited by 548
Cited by3
Results whose statement or proof uses this declaration.
- hasSum_mellinproof · cited by 1
- Complex.tendsto_partialGammaproof · cited by 1
- Complex.Gamma_mul_Gamma_eq_betaIntegralproof · cited by 1