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.