Morera theorem

2020 Mathematics Subject Classification: Primary: 30-XX Secondary: 32-XX [MSN][ZBL]

A fundamental theorem in complex analysis first proved by G. Morera in [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 contour $\gamma\subset D$, then the function $f$ is holomorphic.

The integral in $\eqref{e:integral}$ must be understood in the sense of the usual integration of a $1$-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

[Co]	J.B. Conway, "Functions of one complex variable" , Springer (1973) MR0447532 Zbl 0277.30001
[Re]	R. Remmert, "Funktionentheorie" , 1 , Springer (1984)