So far we have seen a few ways to analytically continue a holomorphic function.
- By integration using theresidue theorem (Analytically continued gamma and zeta function)
- Using the (generalized) Schwarz reflection principal, which we used to define maps from the halfplane onto polygons
Now, we look at a more natural way to extend a holomorphic function. We have already seen that every holomorphic function has a power series expansion at a point, that is, it is locally analytic. Now we look at what happens when we take a point in our radius of convergence, and expand the power series around that point. Can we extend our domain that way?