I've already proved that $$\Gamma (n)= (n-1)!$$ but I don´t really know what else to do to verify that $\Gamma$ is an extension of the factorial function for real numbers (positive) Thank you! And I´m sorry for my language, I am Spanish, so thank you again for trying to understand me.
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$\begingroup$ But there are thousands of possible functions then. I mean, I don´t know how gamma should behave when x is not natural... it doesn´t matter? $\endgroup$– CheshireOct 25, 2013 at 1:13
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$\begingroup$ es.wikipedia.org/wiki/Teorema_de_Bohr-Mollerup $\endgroup$– Will JagyOct 25, 2013 at 1:56
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$\begingroup$ es.wikipedia.org/wiki/Convexidad_logar%C3%ADtmica $\endgroup$– Will JagyOct 25, 2013 at 1:57
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$\begingroup$ es.wikipedia.org/wiki/Funci%C3%B3n_gamma $\endgroup$– Will JagyOct 25, 2013 at 1:58
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2$\begingroup$ Have you looked into the functional equation for $\Gamma (z)$? Its behavior is similar to the factorial function. The technique of analytic continuation is extremely useful for the extension you seek. $\endgroup$– 1233dfvOct 25, 2013 at 19:35
2 Answers
Consider $$\Gamma (z)=\int_0^\infty e^{-t}t^{z}{dt\over t}.$$ We can rewrite this as $$\Gamma (z)=\int_0^1 e^{-t}t^{z}{dt\over t}+\int_1^\infty e^{-t}t^{z}{dt\over t}.$$ In the first term of this sum we see that the power series representation of $e^{-t}$ converges uniformly which implies that the series can be integrated term by term. So, $$\Gamma (z)=\int_0^1\sum_{n=0}^\infty {(-1)^n\over n!}t^{z+n}{dt\over t}+\int_1^\infty e^{-t}t^{z}{dt\over t}=\sum_{n=0}^\infty {(-1)^n\over n!(z+n)}+\int_1^\infty e^{-t}t^{z}{dt\over t}.$$ We can see that the series converges for $z\neq 0, -1, -2,...$ which is a meromorphic function. Its poles are simple poles at the non-positive integers. The residue at $-n$ is $(-1)^n\over n!$. The last integral extends as an entire function of z. Thus $\Gamma (z)$ has been analytically continued to the entire complex plane except for $z\neq0,1,2,...$.
It is a result in Ahlfohrs, Complex Analysis, actually exercise 1 on page 196, that the factorial function can be extended in any way we like at a few non-integer points, and an entire holomorphic function can be constructed to fit those points. In particular, if real valued, we can make any real-analytic extension that we want. So, the fact that the gamma function is analytic is not the big restriction. It is a simpler:
this is the only log-convex extension of the factorial.
http://en.wikipedia.org/wiki/Logarithmic_convexity