 Gamma integral

ID: gamma-integral

For

a, s > 0

, the change of variable

u = a t

in the defining integral for the Gamma function gives

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

In particular,

Γ (1/2) = π

implies

u^{- 1/2} = \frac{1}{π} \int_{0}^{\infty} t^{- 1/2} e^{- u t} d t .

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