# Different Statements of Morera's Theorem

• variety
In summary, the first statement of the theorem states that if a continuous function is analytic on a region, then the integral of the function over any closed path in the region is zero. The second statement of the theorem states that if a continuous function is analytic on a region, then the integral of the function over any simple closed path in the region is also zero.

#### variety

In my Complex Analysis text, Morera's theorem states that a continious function whose domain is a region G is analytic if $$\int_T f=0$$ for every triangular path T in G. However, other versions of the theorem state that the integral must be zero for any simple closed curve in G. Can someone explain how one can go from the first statement to the second one? It doesn't have to be a full proof; just a heuristic argument. I'm just curious. It's probably something to do with the fact that any curve can be segmented into straight line segments or something, but I haven't taken any topology courses so I don't even really know what I'm taking about.

yeah, you've got the idea. these triangular sections will become a more and more fine approximation of the enclosed region, so the first implies the second in the limit. but you really need an oriented manifold for this loop to be in, so that an orientated clockwise ABC triangle will attach to an oriented clockwise DCB triangle to make an oriented ABDC quadralateral. so the idea is to fix an origin and cut up the loop into same-oriented triangles which include the origin as a vertex, and all the interior segments will cancel out, leaving just the loop to integrate.

Let f be a continuous complex-valued function on the open set G.

Thm M1: If $$\int_T f=0$$ for every triangular path in G, then f is analytic on G.

Thm M2: If $$\int_{\Gamma} f=0$$ for every simple closed path $$\Gamma$$ in G, then f is analytic on G.

variety, you asked how to go from Thm M1 to Thm M2. But if you know Thm M1 is true, isn't Thm M2 a trivial corollary? (If the integral is zero for every simple closed path, then the integral is certainly 0 for every triangular path.)