Definitions

Morera's theorem

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(z),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(b) = int_a^b f(z),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'(z) = f(z).,

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(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 = lim_{nrightarrowinfty} oint_C f_n(z),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(s)=sum_{n=1}^infty frac{1}{n^s}

or the Gamma function

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

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. 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

External links

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