# Morera's theorem

File:Morera's Theorem.png
If the integral along every C is zero, then ƒ is holomorphic on D.

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 simply connected open set D in the complex plane that satisfies

$\oint _{\gamma }f(z)\,dz=0$ for every closed piecewise C1 curve $\gamma$ 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

File:Morera's Theorem Proof.png
The integrals along two paths from a to b are equal, since their difference is the integral along a closed loop.

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.

$F(z)=\int _{\gamma }f(\zeta )\,d\zeta .\,$ $\oint _{\gamma }f(\zeta )\,d\zeta \,+\oint _{\tau ^{-1}}f(\zeta )\,d\zeta \,=\oint _{\gamma \tau ^{-1}}f(\zeta )\,d\zeta \,=0$ And it follows that

$\oint _{\gamma }f(\zeta )\,d\zeta \,=\oint _{\tau }f(\zeta )\,d\zeta .\,$ 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 ƒ1ƒ2, ... is a sequence of holomorphic functions, converging uniformly to a continuous function ƒ on an open disc. By Cauchy's theorem, we know that

$\oint _{C}f_{n}(z)\,dz=0$ for every n, along any closed curve C in the disc. Then the uniform convergence implies that

$\oint _{C}f(z)\,dz=\oint _{C}\lim _{n\to \infty }f_{n}(z)\,dz=\lim _{n\to \infty }\oint _{C}f_{n}(z)\,dz=0$ 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

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$ or the Gamma function

$\Gamma (\alpha )=\int _{0}^{\infty }x^{\alpha -1}e^{-x}\,dx.$ Specifically one shows that

$\oint _{C}\Gamma (\alpha )\,d\alpha =0$ for a suitable closed curve C, by writing

$\oint _{C}\Gamma (\alpha )\,d\alpha =\oint _{C}\int _{0}^{\infty }x^{\alpha -1}e^{-x}\,dx\,d\alpha$ and then using Fubini's theorem to justify changing the order of integration, getting

$\int _{0}^{\infty }\oint _{C}x^{\alpha -1}e^{-x}\,d\alpha \,dx=\int _{0}^{\infty }e^{-x}\oint _{C}x^{\alpha -1}\,d\alpha \,dx.$ Then one uses the analyticity of x ↦ xα−1 to conclude that

$\oint _{C}x^{\alpha -1}\,d\alpha =0,$ 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

$\oint _{\partial T}f(z)\,dz$ 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.