# 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 a continuous, complex-valued function *ƒ* defined on a connected open set *D* in the complex plane that satisfies

for every closed piecewise *C*^{1} curve in *D* must be holomorphic on *D*.

The assumption of Morera's theorem is equivalent to that *ƒ* has an antiderivative 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. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero.

## Proof

There is a relatively elementary proof of the theorem. One constructs an anti-derivative for *ƒ* 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 *z*_{0} in *D*, and for any , let be a piecewise *C*^{1} curve such that and . Then define the function *F* to be

To see that the function is well-defined, suppose is another piecewise *C ^{1}* curve such that and . The curve (i.e. the curve combining with in reverse) is a closed piecewise

*C*

^{1}curve in

*D*. Then,

And it follows that

By continuity of f and the definition of the derivative, we get that F'(z) = f(z). Note that we can apply neither the Fundamental theorem of Calculus nor the mean value theorem since they are only true for real-valued functions.

Since *f* is the derivative of the holomorphic function *F*, it is holomorphic. This completes the proof.

## 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 *ƒ*_{1}, *ƒ*_{2}, ... is a sequence of holomorphic functions, converging uniformly to a continuous function *ƒ* on an open disc. By Cauchy's theorem, we know that

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

for every closed curve *C*, and therefore by Morera's theorem *ƒ* 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 in conjunction with Fubini's theorem and the Weierstrass M-test to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function

or the Gamma function

Specifically one shows that

for a suitable closed curve *C*, by writing

and then using Fubini's theorem to justify changing the order of integration, getting

Then one uses the analyticity of *x* ↦ *x*^{α−1} to conclude that

and hence the double integral above is 0. Similarly, in the case of the zeta function, the M-test justifies interchanging the integral along the closed curve and the sum.

## Weakening of hypotheses

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

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

## See also

- Cauchy–Riemann equations
- Methods of contour integration
- Residue (complex analysis)
- Mittag-Leffler's theorem

## References

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