Morera's theorem

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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

\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[edit]

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 z\in D, let \gamma: [0,1]\to D be a piecewise C1 curve such that \gamma(0)=z_0 and \gamma(1)=z. Then define the function F to be

F(z) = \int_\gamma f(\zeta)\,d\zeta.\,

To see that the function is well-defined, suppose \tau: [0,1]\to D is another piecewise C1 curve such that \tau(0)=z_0 and \tau(1)=z. The curve \gamma \tau^{-1} (i.e. the curve combining \gamma with \tau in reverse) is a closed piecewise C1 curve in D. Then,

\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[edit]

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[edit]

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[edit]

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[edit]

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.

See also[edit]

References[edit]

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