Functional equation of the Riemann zeta function

The completed zeta functionis an entire function satisfying $\xi (s)=\xi (1-s)$. Thus every nontrivial zero $\rho$ is accompanied by $1-\rho$, $\overline{\rho }$, and $1-\overline{\rho }$.

For $\Theta (u)={\sum }_{n\in \mathbb{Z}}{e}^{-\pi n^{2}u}$ and initially $\Re s>1$,The Poisson summation formula gives $\Theta (u)=u^{-1/2}\Theta (1/u)$. Splitting the integral at one therefore continues it meromorphically and makes its invariance under $s\mapsto 1-s$ visible.

Let $J(s)={\int }_1^{\infty }(\theta (t)-1)t^{s/2-1}dt$, with $\theta (t)={\sum }_{n\in \mathbb{Z}}{e}^{-\pi n^{2}t}$. Exponential decay makes $J$ an entire function. The Jacobi theta function transformation and a substitution in the Mellin transform giveThis continues the completed Riemann zeta function meromorphically, with residues $1$ and $-1$ at $1$ and $0$, and makes the functional equation of the Riemann zeta function immediate. Dividing by the Gamma function removes the apparent pole at zero from $\zeta (s)$.

Writingthe Stirling formula gives, uniformly on every fixed vertical strip and for large $|t|$,