Morera's theorem

The Morera's theorem , named after Giacinto Morera , is a set of the function theory , a branch of mathematics . Function theory deals with complex differentiable functions and their properties.

If it is open and a function, then it is called holomorphic if it is differentiable from complex in every point . This is a very strong property, for example a holomorphic function is also analytical at the same time , i.e. H. locally developable into a power series. So there are relatively few functions that function theory deals with. Therefore, among other things, very strong conclusions follow from the relatively low assumptions. Morera's theorem allows some such conclusions. ${\displaystyle U\subseteq \mathbb {C} }$ ${\displaystyle f\colon U\to \mathbb {C} }$ ${\displaystyle U}$

The sentence

Several versions of the sentence are common:

1st version

It is open and a continuous function. For every triangle located in the curve integral vanishes over the edge curve of the triangle, i.e. H. . Then holomorphic is on . ${\displaystyle U\subset \mathbb {C} }$ ${\displaystyle f\colon U\rightarrow \mathbb {C} }$ ${\displaystyle U}$ ${\displaystyle \Delta }$ ${\displaystyle \oint _{\partial \Delta }f(z)\,\mathrm {d} z=0}$ ${\displaystyle f}$ ${\displaystyle U}$

2nd version

It is open and a function. If is locally integrable, i. H. if at every point of has a local antiderivative, then is holomorphic on . ${\displaystyle U\subset \mathbb {C} }$ ${\displaystyle f\colon U\rightarrow \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle f}$ ${\displaystyle U}$

In fact, all statements are equivalent:

• 1st version: If is holomorphic, then the curve integral vanishes over the boundary curve of every triangle located in , according to Goursat's lemma . ${\displaystyle f}$ ${\displaystyle U}$
• 2nd version: Since it is open, there is a convex environment for every point . Since is holomorphic, there is an antiderivative of according to Cauchy's integral theorem . ${\displaystyle U}$ ${\displaystyle z\in U}$ ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle B}$ ${\displaystyle f}$

literature

• Eberhard Freitag, Rolf Busam: Function Theory , Springer, Berlin 2000, ISBN 3-540-67641-4