Is Morera's Theorem the Converse of Cauchy's Theorem in Complex Analysis?

[y.moghadamnia]

hey there
there is this thing we learn in complex analysis (and almost everywhere) that if a function is analytic in a known region, then the integral on a closed path(say, any loop), will be zero.
so there is another statement we need to deal with hear, which is exactly the opposite, that if the integral on any closed path is zero, then our function will be analytic. its called morera sth. now, I get it completely, but I have problems with provinging it, can anyone prove it and explain it completely? I tried some textbooks but none had the explanation I truly needed to understand the whole thing.
thanx a lot in advance

 

[Gib Z]

The first statement is called Cauchy's theorem, and at first sight Morera's theorem seems like it's converse, but Cauchy's theorem actually makes an additional assumption that the domain is simply connected.

The proof on Wikipedia ( http://en.wikipedia.org/wiki/Morera's_theorem ) seems straight forward enough. Read through it and if you have problems, come back and tell us specifically what part you have having trouble understanding.