The-gamma-function-s-0-e-x-x-s-1-dx-How-to-calculate-the-gamma-function-in-an-easy-and-not-time-consuming-way-

Question Number 9233 by geovane10math last updated on 24/Nov/16

The gamma function

Γ (s) = \int_{0}^{\infty} e^{- x} x^{s - 1} d x

How to calculate the gamma function

in an easy and not time - consuming

way ?

Answered by 123456 last updated on 25/Nov/16

Γ (s + 1) = s!

you can use stirling aproxiation for

large s, also any good aproximation

can be made by

Γ (s + 1) = s Γ (s) \Leftrightarrow Γ (s) = \frac{Γ (s + 1)}{s}

1 : for ∣ s ∣> ϵ (ϵ large enough)

get aproximate value via stirling

2 : for ∣ s ∣⩽ ϵ

get value via Γ (s) = \frac{Γ (s + 1)}{s}

- - - - - - - - - - - - - - - -

not sure if it eficient but it work : v

Commented by 123456 last updated on 25/Nov/16

Γ (s) = \frac{Γ (s + 1)}{s}

Γ (s + 1) = \frac{Γ (s + 2)}{s + 1} \Rightarrow Γ (s) = \frac{Γ (s + 2)}{s (s + 1)}

\dots

Γ (s) = \frac{Γ (s + a)}{s (s + 1) \dots (s + a - 1)} a \in N, a > 0

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