Next: The theorems of Liouville Up: Integrals. Previous: Generalization of Cauchy's Integral Contents 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 CauchyGoursat 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 CauchyGoursat theorem, stronger than Prop. 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 Next: The theorems of Liouville Up: Integrals. Previous: Generalization of Cauchy's Integral Contents Noah DanaPicard 20040126  
