Cauchy’s Theorem

Theorem (Cauchy’s Theorem (Homotopy Version))

Let be a connected open subset of the complex plane. Let be a holomorphic function. If , are homotopic closed curves in , then

If is a null-homotopic closed curve in , then

Cauchy’s Theorem has a converse:

Morera’s Theorem

Let be open and let be a continuous function. If for every contour contained in , then is holomorphic in .