Morera's theorem

The set of Morera, named after Giacinto Morera, is a set of function theory, a branch of mathematics. The function theory deals with complex- differentiable functions and their properties.

Is open and a function, then it is called holomorphic if it is complex differentiable at every point of. This is a very powerful property, such as a holomorphic function is analytic at the same time, that is locally developable in a power series. So there are relatively few features that enable the function theory is concerned. One of the reasons follow from fairly low requirements very strong conclusions. Some of these circuits allows the set of Morera.

The set

There are several versions of the sentence usual:

First version

It was open and a continuous function. For each location in the triangle disappear the line integral over the boundary curve of the triangle, ie. Then is holomorphic on.

2nd version

It was open and a function. If is locally integrable, that is, if at any point of having a local root function, then is holomorphic on.

In fact, all statements are equivalent: