Definitions

# Morera's theorem

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

Morera's theorem states that if f is a continuous, complex-valued function defined on an open set D in the complex plane, satisfying

$oint_C f\left(z\right),dz = 0$

for every closed curve C in D, then f must be holomorphic on D.

The assumption of Morera's theorem is equivalent to that f has an anti-derivative on D.

The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. For instance, Cauchy's integral theorem states that the line integral of a holomorphic function along a closed curve is zero, provided that the domain of the function is simply connected.

## Proof

There is a relatively elementary proof of the theorem. One constructs an anti-derivative for f explicitly. The theorem then follows from the fact that holomorphic functions are analytic.

Without loss of generality, it can be assumed that D is connected. Fix a point a in D, and define a complex-valued function F on D by

$F\left(b\right) = int_a^b f\left(z\right),dz.,$

The integral above may be taken over any path in D from a to b. The function F is well-defined because, by hypothesis, the integral of f along any two curves from a to b must be equal. It follows from the fundamental theorem of calculus that the derivative of F is f:

$F\text{'}\left(z\right) = f\left(z\right).,$

In particular, the function F is holomorphic. Then f must be holomorphic as well, being the derivative of a holomorphic function.

## Applications

Morera's theorem is a standard tool in complex analysis. It is used in almost any argument that involves a non-algebraic construction of a holomorphic function.

### Uniform limits

For example, suppose that f1f2, ... is a sequence of holomorphic functions, converging uniformly to a continuous function f on an open disc. By Cauchy's theorem, we know that

$oint_C f_n\left(z\right),dz = 0$

for every n, along any closed curve C in the disc. Then the uniform convergence implies that

$oint_C f\left(z\right),dz = lim_\left\{nrightarrowinfty\right\} oint_C f_n\left(z\right),dz = 0$

for every closed curve C, and therefore by Morera's theorem f must be holomorphic. This fact can be used to show that, for any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a Banach space with respect to the supremum norm.

### Infinite sums and integrals

Morera's theorem can also be used to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function

$zeta\left(s\right)=sum_\left\{n=1\right\}^infty frac\left\{1\right\}\left\{n^s\right\}$

or the Gamma function

$Gamma\left(alpha\right)=int_0^infty x^\left\{alpha-1\right\} e^\left\{-x\right\},dx.$

## Weakening of hypotheses

The hypotheses of Morera's theorem can be weakened considerably. In particular, it suffices for the integral

$oint_\left\{partial T\right\} f\left(z\right), dz$

to be zero for every closed triangle T contained in the region D. This in fact characterizes holomorphy, i.e. f is holomorphic on D if and only if the above conditions hold.

## References

• G. Morera, "Un teorema fondamentale nella teoria delle funzioni di una variabile complessa", Rend. del R. Instituto Lombardo di Scienze e Lettere (2) 19 (1886) 304–307