# Functional equation for the zeta function

The **functional equation for Riemann zeta function** is a result due to analytic continuation of Riemann zeta function:

## Contents

## Proof

### Preparation

There are multiple proofs for the functional equation for Riemann zeta function, and this page presents a light-weighted approach which merely relies on the Fourier series for the first periodic Bernoulli polynomial that

and the Laplace transform identity that

where

### A formula for in

In this article, we will use the common convention that where . As a result, we say that the original Dirichlet series definition converges only for . However, if we were to apply Euler-Maclaurin summation on this definition, we obtain

in which we can extend the ROC of the latter integral to via integration by parts:

When there is

As a result, we obtain a formula for for :

### Expansion of into Fourier series

In order to go deeper, let's plug

into the previously obtained formula, so that

Therefore, the remaining step is to handle the integral

### Evaluation of

By Euler's formula, we have

As a result, we only need to calculate

if we want to take down the remaining integral. According to Laplace transform identities, we can see that

Thus we deduce

wherein the RHS serves to be a meromorphic continuation of the LHS integral.

### Proof of the functional equation

With everything ready, we can put everything together and obtain

and by , this identity becomes the functional equation:

## Resources

- Titchmarsh, E. C.,
*The Theory of the Riemann Zeta-Function.*Oxford Univ. Press, London and New York, 1951.