# 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 connected open set D in the complex plane that satisfies

${\displaystyle \oint _{\gamma }f(z)\,dz=0}$

for every closed piecewise C1 curve ${\displaystyle \gamma }$ 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.

Without loss of generality, it can be assumed that D is connected. Fix a point z0 in D, and for any ${\displaystyle z\in D}$, let ${\displaystyle \gamma :[0,1]\to D}$ be a piecewise C1 curve such that ${\displaystyle \gamma (0)=z_{0}}$ and ${\displaystyle \gamma (1)=z}$. Then define the function F to be

${\displaystyle F(z)=\int _{\gamma }f(\zeta )\,d\zeta .\,}$

To see that the function is well-defined, suppose ${\displaystyle \tau :[0,1]\to D}$ is another piecewise C1 curve such that ${\displaystyle \tau (0)=z_{0}}$ and ${\displaystyle \tau (1)=z}$. The curve ${\displaystyle \gamma \tau ^{-1}}$ (i.e. the curve combining ${\displaystyle \gamma }$ with ${\displaystyle \tau }$ in reverse) is a closed piecewise C1 curve in D. Then,

${\displaystyle \int _{\gamma }f(\zeta )\,d\zeta \,+\int _{\tau ^{-1}}f(\zeta )\,d\zeta \,=\oint _{\gamma \tau ^{-1}}f(\zeta )\,d\zeta \,=0}$

And it follows that

${\displaystyle \int _{\gamma }f(\zeta )\,d\zeta \,=\int _{\tau }f(\zeta )\,d\zeta .\,}$

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

${\displaystyle \oint _{C}f_{n}(z)\,dz=0}$

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

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

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

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}}$

or the Gamma function

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

Specifically one shows that

${\displaystyle \oint _{C}\Gamma (\alpha )\,d\alpha =0}$

for a suitable closed curve C, by writing

${\displaystyle \oint _{C}\Gamma (\alpha )\,d\alpha =\oint _{C}\int _{0}^{\infty }x^{\alpha -1}e^{-x}\,dx\,d\alpha }$

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

${\displaystyle \int _{0}^{\infty }\oint _{C}x^{\alpha -1}e^{-x}\,d\alpha \,dx=\int _{0}^{\infty }e^{-x}\oint _{C}x^{\alpha -1}\,d\alpha \,dx.}$

Then one uses the analyticity of x ↦ xα−1 to conclude that

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

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

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

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.

## References

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