Morera's theorem

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If the integral along every C is zero, then f is holomorphic on D.
If the integral along every C is zero, then f 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 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.

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[edit] Proof

The integrals along two paths from a to b are equal, since their difference is the integral along a closed loop.
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 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.

[edit] 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.

[edit] Uniform limits

For example, suppose that f1, f2, ... 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_{n\rightarrow\infty} \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.

[edit] 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.

[edit] 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.

[edit] References

  • Ahlfors, Lars (January 1, 1979), Complex Analysis, McGraw-Hill, ISBN 978-0070006577 
  • Conway, John B. (April 1, 2001), Functions of One Complex Variable I, Graduate Texts in Mathematics, Springer, ISBN 978-3540903284 
  • 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
  • Rudin, Walter (May 1, 1986), Real and Complex Analysis, McGraw-Hill, ISBN 978-0070542341 

[edit] External links