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 August 20th, 2014, 03:19 PM #1 Senior Member   Joined: Aug 2014 From: United States Posts: 137 Thanks: 21 Math Focus: Learning Understanding Analytic Continuation Can somebody help me to further increase my understanding of analytic continuation and its applications? At this point I understand that analytic continuation is a method of increasing the domain of a complex function. I also know that if two functions, say $f$ and $g$, are analytic on two domains, say $\Omega_f$ and $\Omega_g$ respectively, where $f=g$ on the domain $\Omega_f\cap\Omega_g$, then $g$ is an analytic continuation of $f$. Is this correct? And also, what is this used for, and what's it got to do with the Riemann Zeta Function?
 August 20th, 2014, 06:23 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra Have you googled? WA, wikipedia both have articles that give more than you wrote. Thanks from neelmodi
August 21st, 2014, 03:52 PM   #3
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 Originally Posted by neelmodi Can somebody help me to further increase my understanding of analytic continuation and its applications? At this point I understand that analytic continuation is a method of increasing the domain of a complex function. I also know that if two functions, say $f$ and $g$, are analytic on two domains, say $\Omega_f$ and $\Omega_g$ respectively, where $f=g$ on the domain $\Omega_f\cap\Omega_g$, then $g$ is an analytic continuation of $f$. Is this correct? And also, what is this used for, and what's it got to do with the Riemann Zeta Function?
You need some restriction on $\Omega_f\cap\Omega_g$, such as containing an open set.

The infinite series for the Zeta function converges only for Re(s) > 1. Continuation allows it to be extended to the whole complex plane, except for pole at s=1.

Riemann zeta function - Wikipedia, the free encyclopedia

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