split into two integrals over two different closed contours: one that lies in the upper half plane and one in the lower. by the Cauchy theorem each of the integrals equals 0.
is continuous on and analytic on .
Prove that is an entire function.
I know that I need to apply Morera's Theorem which states that if is a continuous, complex-valued function defined on an open set , satisfying
for every closed curve in , then must be holomorphic on .
I guess I'm not sure how to show that satisfies . I thought I could use Cauchy's integral theorem to show that the function does satisfy this condition but I only know that is holomorphic on . Any help would be much appreciated.
Thank you for your help. I can see how to apply Cauchy's integral theorem in this manner, but in conjunction with Morera's theorem it seems as though I am only showing that is holomorphic on but in order to show that is entire I must show it is holomorphic on Is there something I am missing here?