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 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 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 Dana-Picard 2004-01-26 | |
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