Functional Equation for Riemann Zeta Function

Theorem
Let $\zeta$ be the Riemann zeta function.

Let $\map \zeta s$ have an analytic continuation for $\map \Re s > 0$.

Then:
 * $\ds \pi^{-s/2 } \map \Gamma {\dfrac s 2} \map \zeta s = \pi^{\paren {s/2 - 1/2 } } \map \Gamma {\dfrac {1 - s} 2} \map \zeta {1 - s}$

where $\Gamma$ is the gamma function

Proof
Let $\ds \map \omega x = \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x}$.

Then from Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function we have:


 * $(1): \quad \ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = -\frac 1 {s \paren {1 - s} } + \int_1^\infty \paren {x^{s / 2 - 1} + x^{- s / 2 - 1 / 2} } \map \omega x \rd x$

We observe that this integral is invariant under $s \mapsto 1 - s$.

Then:

as required.

Also see

 * Functional Equation for Completed Riemann Zeta Function


 * Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function