proof of Morera’s theorem
We provide a proof of Morera’s theorem under the hypothesis that for any circuit contained in . This is apparently more restrictive, but actually equivalent, to supposing for any triangle , provided that is continuous in .
The idea is to prove that has an antiderivative in . Then , being holomorphic in , will have derivatives of any order in ; but for all , , .
Fix . For any define as
where is a path entirely contained in with initial point and final point .
The function is well defined. In fact, let and be any two paths entirely contained in with initial point and final point ; define a circuit by joining and , the path obtained from by “reversing the parameter direction”. Then by linearity and additivity of integral
but the left-hand side is 0 by hypothesis, thus the two integrals on the right-hand side are equal.
We must now prove that in . Given , there exists such that the ball of radius centered in is contained in . Suppose : then we can choose as a path from to the segment parameterized by . Write with : by additivity of integral and the mean value theorem,
for some . Since is continuous, so are and , and
In the general case, we just repeat the procedure once for each connected component of .