TITLE: How is extension by analytic continuation done?
QUESTION [5 upvotes]: I understand  that by defining an expression for a function to be analytic, we can extend the range of the expression beyond it's usual range. One case is that a series which is asymptotic to a function defined abstractly within certain domain could in fact diverage outside of that domain while the function is still defined. 
Can analytic continuation enable us to extend the range of an expression from just a little block of known domain and range to infinite range and domain? If so, how is it done?
Could it be only done by computer simulation? Or algebraic manipulation is enough? What are the ways to do this? Could it be done by hand in special case? In general, how is it done?

REPLY [2 votes]: A nice example of analytic continuation is given by the closed-form expression for a  geometric series. Consider 
$$
f(z) = \sum_{n = 0}^\infty z^n.
$$
A priori, this is defined on the open disk $|z| < 1$. However, we know that $g(z)  \frac{1}{1 - z}$ is exactly the function $f(z)$ on this disk, and yet $g$ is defined on all complex numbers except  $z = 1$. Thus, $g$ may be thought of as  an analytic extension of the original function $f$. This extends our original series $f$, originally defined on the open unit disk, to have domain $\mathbb{C} \setminus \{1\}$,  provided that outside the disk, we use the formula for $g$. In particular, note that $f$ does not really make sense  outside of the disk, but $g$ does.  
You could also look up analytic continuation of the gamma function or Riemann-zeta functions.
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