Hi, I see a formula of gamma function and i have a question.(adsbygoogle = window.adsbygoogle || []).push({});

(1) $$\Gamma (s) = \int_{0}^{\infty } e^{-x}\, x^{s-1} dx$$

(2) $$ x=a\, n^{p} \rightarrow dx=ap\, n^{p-1}dn$$

(3) $$\frac{\Gamma (s)}{pa^{s}} = \int_{0}^{\infty } e^{-an^{p}}\, n^{ps-1} dn$$

i understand the formula but why works for complex values of a?

i mean the substitution in (2) is valid for ##a > 0## and ##p>0##

Because otherwise the upper limit would be ##\pm \infty \, i ## or ## \infty \ \pm \infty \, i ##

Depending on the value of a.

thanks

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# I Question about Complex limits of definite integrals

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