# Morera's Theorem.

In two variable Calculus, you learnt the following result: let and be two functions defined on the same simply connected domain . Suppose that and are continuous on the interior of and that, for every Jordan curve in , the following equation holds:

Then in .

Using this result we can prove the following converse to Cauchy-Goursat theorem:

Proposition 5.6.1   Let be a function such that are continuous in a simply connected domain . Suppose that, for every Jordan curve in , the integral is equal to 0. Then is analytic on .

Another converse of Cauchy-Goursat theorem, stronger than Proposition 6.1 is Morera's theorem:

Theorem 5.6.2 (Morera)   Let be a continuous function on an open simply connected domain . Assume that for every loop in , the integral is equal to 0. Then is analytic on .

Example 5.6.3

Noah Dana-Picard 2007-12-24