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1) Firstly, why would you want to apply analytic continuation on some complex function?

2) Surely if you apply analytic continuation on some complex function, then it's no longer the same function?

3) What is analytic continuation, really?

I came across this concept when I was reading about the Riemann hypothesis in some popular articles and books. They mentioned that the Riemann zeta function is the infinite sum of [tex]1/n^s[/tex] over natural numbers, n, but this is only the case when the real part of s is greater than 1; values of the Riemann zeta function for the rest of the complex plane may be found via analytic continuation.

What kind of book would give me an introduction to analytic continuation? A complex analysis book? Any recommendations? Also, what sort of mathematical background do I need to be able to understand one of the standard books devoted to the Riemann zeta function, like "Riemann's Zeta Function" by Harold M. Edwards, or "The Theory of the Riemann Zeta-Function" by E. C. Titchmarsh?