Theorems · Theorem · special functions
Real.GammaIntegral_convergent
∀ {s : ℝ}, 0 < s → MeasureTheory.IntegrableOn (fun x => Real.exp (-x) * x ^ (s - 1)) (Set.Ioi 0) MeasureTheory.volumeThe Euler integral for the Γ function converges for positive real s.
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 271 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Realstatement and proof · cited by 25,697
- Nat.cast_oneproof · cited by 2,501
- Nat.cast_zeroproof · cited by 1,870
- Set.Ioistatement · cited by 1,463
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
- Real.expstatement and proof · cited by 871
- MeasureTheory.IntegrableOnstatement and proof · cited by 548
- lt_of_not_geproof · cited by 374
- zero_le_oneproof · cited by 316
- enorm_ne_topproof · cited by 82
- Asymptotics.IsLittleO.isBigOproof · cited by 35
Cited by5
Results whose statement or proof uses this declaration.
- Real.Gamma_pos_of_posproof · cited by 17
- Complex.GammaIntegral_convergentproof · cited by 3
- Complex.approx_Gamma_integral_tendsto_Gamma_integralproof · cited by 1
- Real.Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gammaproof · cited by 1
- integrableOn_rpow_mul_exp_neg_rpowproof · cited by 1