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Let us consider the famous gamma function.

My question is:

Is it possible to derive a closed expression for the gamma function for complex numbers $s=a+ib$ with $a≠1/2$, $b≠0$.

I know that such a closed form exists for integer values of $x=n$ for which $G(n)=(n-1)!$. Also, some non integers values have a closed form. Also if $a=1/2$ then a closed form exists. See this link:

https://mathoverflow.net/questions/112682/riemann-siegel-function-and-gamma-function

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    Thank you very much. This would help.2012-12-23

1 Answers 1

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The $\Gamma$ function lacks a closed form containing only elementary functions. Here are the equivalent formulas of the $\Gamma$ function however:

$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$ $\Gamma(z)=\frac1z\prod_{n=1}^{\infty}\frac{(1+\frac1n)^z}{1+\frac zn}$ $\Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{n=1}^{\infty}\frac{e^{\frac zn}}{1+\frac zn}$ where $\gamma$ is the Euler–Mascheroni constant. Of course we can relate the $\Gamma$ function with elementary functions via the following indentity: $\Gamma(1-z)\Gamma(z)=\frac{\pi}{\sin \pi z}$ We also have Riemann's functional equation $\zeta(s)=2^s\pi^{s-1}\sin\frac{\pi s}2\Gamma(1-s)\zeta(1-s)$ which relates the $\Gamma$ and the $\zeta$ functions.

If by closed form we mean an expression containing only elementary functions then no, $\Gamma$ has no such form. For more information read this

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    I think the last link is dead... :(2018-09-27