In this post,one of the answers (in fact the answer with more upvotes) uses euler's reflection formula $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi\,z)}$ for the gamma function $\Gamma(z)$ to evaluate the expression $\Gamma(\frac{1}{2})$.But in the comments it is said that the formula is much more advanced than evaluating $\Gamma(\frac{1}{2})$.The question that sprang to my mind when I read that comment was "whether this comment was true in general".

Out of curiosity I'd like to know whether the comment is true in general or just a matter of opinion.Possible examples of where the reflection formula is applied except for evaluating the aforementioned expression will be appreciated.

  • $\begingroup$ which statement exactly do you mean? $\endgroup$ – tired Sep 24 '15 at 10:41
  • $\begingroup$ @tired:the fact that "euler's formula is much more advanced than evaluating $\Gamma(\frac{1}{2})$",that is found in the comments. $\endgroup$ – Nicco Sep 24 '15 at 10:45
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    $\begingroup$ It comes up in the functional equation for the Riemann zeta function, which in turn is used to provide the analytic continuation of the zeta function from just being defined for real part exceeding 1 to the entire complex plane (excluding $z=1$). $\endgroup$ – Gerry Myerson Sep 24 '15 at 10:54

The most useful application of this formula is in my opinion the fact that it yields an analytic continuation of the $\Gamma$-function into the the left half of the complex plane $\{z\in\mathbb{C},\Re(z)<0\}$ , excluding the poles at $z_n=-n$ of course. This fact has wide applications, like the continuation of $\zeta(s)$ as mentioned by @Gerry Myerson in the comments.

Furthermore it simplifies some integrals nicely which are related to the Beta function: $$ \int_0^1 t^{z-1}(1-t)^z=B(z,1-z)=\frac{\Gamma(z)\Gamma(1-z)}{\Gamma(1)}=\frac{\pi}{\sin(\pi z )} $$

Which i find particulary cute

To answer the first part of your question: I think it depends, if you already know the reflection formula then this is the easy way to prove $\Gamma(\frac{1}{2})=\sqrt{\pi}$. If you don't, there are indeed more straightforward ways, which u can read up in the linked topic.

  • $\begingroup$ @ tired:fantastic applications indeed.are there any other applications besides the ones you've already mentioned? $\endgroup$ – Nicco Sep 24 '15 at 11:42
  • $\begingroup$ @Nicco you can use this relation also to proof the partial fraction decomposition of trigonometric functions. $\endgroup$ – tired Sep 24 '15 at 12:16
  • $\begingroup$ @ tired:thank you very much $\endgroup$ – Nicco Sep 24 '15 at 12:32

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