Gamma function limit to integral question

Asked Nov 19 '18 at 08:19

Active Nov 27 '19 at 15:53

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I am reading up on the gamma function and have seen a formula that I can't connect to the usual integral definition. Namely,

$$ \Gamma(x) = \lim_{n\rightarrow \infty}\frac{n!n^{x-1}}{x(x+1)\cdots(x+n-1)}, \qquad x\neq 0,-1,-2,\dots $$

How can I connect this formula with the standard definition:

$$ \Gamma(x) = \int^\infty_0 t^{x-1}e^{-t}dt $$

edited Nov 27 '19 at 15:53

Shootforthemoon

asked Nov 19 '18 at 08:19

RedPen

This is standard text book material. Any book that deals with Gamma function in detail has a proof. – Kavi Rama Murthy Nov 19 '18 at 08:25
This can be shown using the factorization of $\frac 1{\Gamma(z)}$, which can be derived from the functional equation $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$. – lEm Nov 19 '18 at 08:35

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