Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$. I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. 
 $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$$
There are probably countless proofs of Riemann's functional equation, but as of yet there's not a single place where they're all concentrated. 
Anyone care to share some particularly pretty proofs?

$\zeta(s)$: Riemann zeta function.
$\Gamma(s)$: Gamma function.
 A: AFAIK none of proofs is very short and easy. I'll post just a rough sketch (in particular all analytical issues are silently ignored) of a Riemann's original proof based on the Poisson summation formula.
Let's define
$$
\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
Riemann's functional equation takes the form
$$
\xi(s)=\xi(1-s).
$$
By definition
$$
\xi(s)=
\sum_{n=1}^\infty\pi^{-s/2}n^{-s}\int_0^\infty e^{-t}t^{s/2}\frac{dt}t
$$
and after substitution $t=\pi n^2x$ we get
$$
\xi(s)=
\int_0^\infty x^{s/2}\sum_{n=1}^\infty e^{-\pi n^2x}\frac{dx}x=
\frac12\int_0^\infty x^{s/2}(\theta(x)-1)\frac{dx}x,
$$
where $\theta(x)=\sum_{n\in\mathbb Z}e^{-\pi n^2x}$.
By Poisson summation formula this theta function satisfies
$$
\theta(x^{-1})=x^{1/2}\theta(x),
$$
which allows us to rewrite last integral in the form
\begin{multline}
\xi(s)=
\frac12\int_1^\infty x^{s/2}(\theta(x)-1)\frac{dx}x+
\frac12\int_1^\infty x^{-s/2}(x^{1/2}\theta(x)-1)\frac{dx}x=\\
=\frac12\int_1^\infty(x^{s/2}+x^{(1-s)/2})(\theta(x)-1)\frac{dx}x-\frac1{s(1-s)},
\end{multline}
which is manifestly symmetric under change $s\mapsto1-s$.

Summary:


*

*The functional equation takes simpler form for the $\xi$-function.

*The $\xi$-function is (more or less) the Mellin transform of $\theta$-function.

*The $\theta$-function satisfies some modularity functional equation coming from the Poisson summation formula.


...or to make a long story short,

Riemann's functional equation is the Mellin transform of the Poisson summation formula.

