Morera's theorem

# Morera's theorem

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Encyclopedia

In complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Morera's theorem, named after Giacinto Morera
Giacinto Morera
Giacinto Morera , was an Italian engineer and mathematician. He is remembered for Morera's theorem in the theory of functions of a complex variables and for his work in the theory of linear elasticity....

, gives an important criterion for proving that a function is holomorphic
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

.

Morera's theorem states that a continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

, complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

-valued function ƒ defined on a connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

D in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

that satisfies

for every closed piecewise C1 curve in D must be holomorphic on D.

The assumption of Morera's theorem is equivalent to that ƒ has an anti-derivative on D.

The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative
Antiderivative (complex analysis)
In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g...

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
Cauchy's integral theorem
In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane...

, stating that the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

of a holomorphic function along a closed curve is zero.

## Proof

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
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

. Fix a point z0 in D, and for any , let be a piecewise C1 curve such that and . Then define the function F to be

To see that the function is well-defined, suppose is another piecewise C1 curve such that and . The curve (i.e. the curve combining with in reverse) is a closed piecewise C1 curve in D. Then,

And it follows that

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
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

. 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
Cauchy's integral theorem
In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane...

, we know that

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

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
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

Ω ⊆ C, the set A(Ω) of all bounded
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...

, analytic functions u : Ω → C is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

with respect to the supremum norm.

### Infinite sums and integrals

Morera's theorem can also be used in conjunction with Fubini's theorem
Fubini's theorem
In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...

to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function

or the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

Specifically one shows that

for a suitable closed curve C, by writing

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

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

and hence the double integral above is 0. Similarly, in the case of the zeta function, Fubini's theorem 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

to be zero for every closed triangle T contained in the region D. This in fact characterizes
Characterization (mathematics)
In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...

holomorphy, i.e. ƒ is holomorphic on D if and only if the above conditions hold.