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let-x-0-t-x-1-e-t-dt-with-x-gt-1-calculate-n-x-for-all-integr-n-

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Question Number 62416 by mathmax by abdo last updated on 20/Jun/19
let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>1 calculate Γ^((n)) (x) for all integr n.
letΓ(x)=∫0∞tx−1e−tdtwithx>1calculateΓ(n)(x)forallintegrn.
Commented by mathmax by abdo last updated on 23/Jun/19
the function Γ is C^∞ on ]0,+∞[ we have Γ(x) =∫_0 ^∞ e^((x−1)ln(t)) e^(−t) dt ⇒ Γ^((1)) (x) =∫_0 ^∞ ln(t) t^(x−1) e^(−t) dt and by recurence we get Γ^((n)) (x) =∫_0 ^∞ (ln(t))^n t^(x−1) e^(−t) dt ∀ n ≥1 .
thefunctionΓisC∞on]0,+∞[wehaveΓ(x)=∫0∞e(x−1)ln(t)e−tdt⇒Γ(1)(x)=∫0∞ln(t)tx−1e−tdtandbyrecurencewegetΓ(n)(x)=∫0∞(ln(t))ntx−1e−tdt∀n⩾1.

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