What is the point of analytic continuation? Surely if a function is not valid in some of the complex plane and you change it so it becomes valid, you've created a new function, telling you nothing about the original one.
Uh? Take for example the series $\displaystyle \displaystyle{1+z+z^2+...=\sum\limits^\infty_{n=0} z^n}$ . This series converges
and defines an analytic function within the unit disk, but we can continue its definition as $\displaystyle \displaystyle{\frac{1}{1-z}}$ to the whole
plane except at the point $\displaystyle z=1$ .
There are other much more important examples, say the zeta Riemann function (see http://numbers.computation.free.fr/C...neralities.pdf , for ex.). Google it.
Tonio