Let , with . Set . It is easy to see that F(s) is defined for s > 0. Consider the new variable r = st. Then we have
,
which implies
,
where the Gamma function is defined by
The domain of the Gamma function is x > 1. Using integration by part, one can easily prove the fundamental formula
,
which implies (knowing that )
Consequently, we have
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Author: Mohamed Amine Khamsi