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 . This series converges
and defines an analytic function within the unit disk, but we can continue its definition as to the whole
plane except at the point .
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.