I'm a high school senior attempting to do a project on the riemann zeta function. I've looked online, tried reading college textbooks but still don't have a completely clear idea of analytic continuation.
Can you have an analytic continuation for all functions or is it only analytic functions?
http://mathpages.com/home/kmath649/kmath649.htm On this page I saw that analytic function works by extending the domain by means of a taylor expansion to represent the infinite points on a complex plane. As long as you have two functions that emit the same value - is this analytic continuation? Or must you do it as it is on the page.
Appreciate the answers. Apologies if this question isn't coherent.
If $f : U \to \mathbb{C}$ is an analytic function defined on domain $U$, then an analytic continuation of $f$ is simply an extension of $f$ to an analytic function defined on a larger domain. Namely, if $V \supseteq U$ is a domain and $F : V \to \mathbb{C}$ is analytic, it is an analytic continuation of $f$ precisely if its restriction to $U$ is $f$. Note that the extension $F$ must be analytic in order to count as an analytic continuation. Note also that $F$ is uniquely determined when $V$ is connected: if $F$ and $G$ are both analytic continuations of $f$ then on $U$, $F-G = 0$, and so by the identity theorem $F-G = 0$ on $V$. Therefore it makes sense to talk about THE analytic continuation of $f$ to $V$.