View source for Morera theorem

{{MSC|30-XX|32-XX}}
{{TEX|done}}

A fundamental theorem in complex analysis first proved by G. Morera in {{Cite|Mo}}, which is an (incomplete) converse of the [[Cauchy integral theorem]]. The theorem states the following. 

'''Theorem'''
Let $D\subset \mathbb C$ be an open set and $f: D\to \mathbb C$ a continuous function. If
the integral
\begin{equation}\label{e:integral}
\int_\gamma f(z)\, dz = 0
\end{equation}
vanishes for every [[Rectifiable curve|rectifiable]] contour $\gamma\subset D$, then the function $f$ is [[Holomorphic function|holomorphic]].
 
The integral in \eqref{e:integral} must be understood in the sense of the usual [[Integration on manifolds|integration]] of a $1$-[[Differential form|form]]. In particular, if $z: [0,T]\to D$ is a Lipschitz parametrization of the contour $\gamma$, then the right hand side of \eqref{e:integral} is given by
\[
\int_0^T f (z(t))\, \dot{z} (t)\, dt\, .
\]
Indeed the assumption of the theorem can be considerably weakened: to conclude that $f$ is holomorphic it suffices to know \eqref{e:integral} whenever $\gamma$ is the boundary of any triangle $\Delta\subset\subset D$.

Morera's theorem can be generalized to functions of several complex variables. 

'''Theorem'''
Let $D\subset \mathbb C^n$ be an open set and $f: D \to \mathbb C$ a continuous function. Denote by $f (z)\, dz$ the (complex) differential form
\[
f (z)\, dz_1\wedge dz_2\wedge \ldots \wedge dz_n\, .
\]
Consider the class $\mathcal{P}$ of ''prismatic'' domains $\Gamma\subset\subset D$ of the form
\[
[a_1, b_1] \times \ldots \times [a_{i-1}, b_{i-1}]\times \partial \Delta \times [a_{i+1}, b_{i+1}] \times \ldots \times [a_n b_n]\, ,
\]
where $\Delta\subset \mathbb C$ is a arbitrary triangle, $a_k, b_k$ are complex numbers and $[a_k, b_k]$ denotes the segment $\sigma\subset \mathbb C$ given by $\{\lambda a_k + (1-\lambda b_k): \lambda \in [0,1]\}$. If
\[
\int_\Gamma f(z)\, dz = 0\,  \qquad\qquad \mbox{for any}\, \Gamma \in \mathcal{P}\, ,
\]
then $f$ is holomorphic.

====References====
{|
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|valign="top"|{{Ref|Co}}|| J.B. Conway, "Functions of one complex variable" , Springer (1973) {{MR|0447532}} {{ZBL|0277.30001}}
|-
|valign="top"|{{Ref|Re}}|| R. Remmert,   "Funktionentheorie" , '''1''' , Springer  (1984)
|-
|}