I was reading through my text, and it stated that (for some integer n) was not analytic at n = 0. Why is this the case? Usually I'd substitute z = x + iy and then use the Cauchy-Riemann equations to check, but (x + iy)^n makes it difficult to separate the real and imaginary parts.
Technically is undefined, but if we check the limit along any path of
So by the Riemann removable singularity theorem the function can be analytically extended to all of the complex plane
Removable singularity - Wikipedia, the free encyclopedia
Note that holomorphic function is a function that is analytic on all of the complex plane.