Functional equation for the zeta function
Introduction
The functional equation for Riemann zeta function is a result due to analytic continuation of Riemann zeta function:
Proof
Two useful identities
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
From
to ![$\sigma>-1$](//latex.artofproblemsolving.com/a/c/0/ac02eef88cb7357e01994953495b806f68838b18.png)
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 repeated integration: