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In
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 helpful in many branches of mathematics, including algebraic ...
, a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Morera's theorem, named after Giacinto Morera, gives a criterion for proving that a function is holomorphic. Morera's theorem states that a continuous,
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-valued function ''f'' defined on an
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
''D'' in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
that satisfies \oint_\gamma f(z)\,dz = 0 for every closed piecewise ''C''1 curve \gamma in ''D'' must be holomorphic on ''D''. The assumption of Morera's theorem is equivalent to ''f'' having an
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...
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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
; this is Cauchy's integral theorem, stating that the
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...
of a holomorphic function along a
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
is zero. The standard counterexample is the function , which is holomorphic on C − . On any simply connected neighborhood U in C − , 1/''z'' has an antiderivative defined by , where . Because of the ambiguity of ''θ'' up to the addition of any integer multiple of 2, any continuous choice of ''θ'' on ''U'' will suffice to define an antiderivative of 1/''z'' on ''U''. (It is the fact that ''θ'' cannot be defined continuously on a simple closed curve containing the origin in its interior that is the root of why 1/''z'' has no antiderivative on its entire domain C − .) And because the derivative of an additive constant is 0, any constant may be added to the antiderivative and the result will still be an antiderivative of 1/''z''. In a certain sense, the 1/''z'' counterexample is universal: For every analytic function that has no antiderivative on its domain, the reason for this is that 1/''z'' itself does not have an antiderivative on C − .


Proof

There is a relatively elementary proof of the theorem. One constructs an anti-derivative for ''f'' explicitly. Without loss of generality, it can be assumed that ''D'' is connected. Fix a point ''z''0 in ''D'', and for any z\in D, let \gamma: ,1to D be a piecewise ''C''1 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: ,1to D is another piecewise ''C''1 curve such that \tau(0)=z_0 and \tau(1)=z. The curve \gamma \tau^ (i.e. the curve combining \gamma with \tau in reverse) is a closed piecewise ''C''1 curve in ''D''. Then, \int_ f(\zeta)\,d\zeta + \int_ f(\zeta) \, d\zeta =\oint_ f(\zeta)\,d\zeta = 0. And it follows that \int_\gamma f(\zeta)\,d\zeta = \int_\tau f(\zeta)\,d\zeta. Then using the continuity of ''f'' to estimate difference quotients, we get that ''F''′(''z'') = ''f''(''z''). Had we chosen a different ''z''0 in ''D'', ''F'' would change by a constant: namely, the result of integrating ''f'' along ''any'' piecewise regular curve between the new ''z''0 and the old, and this does not change the derivative. Since ''f'' is the derivative of the holomorphic function ''F'', it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that holomorphic functions are analytic, i.e. can be represented by a convergent
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
, and the fact that power series may be differentiated term by term. This completes the proof.


Applications

Morera's theorem is a standard tool in
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 helpful in many branches of mathematics, including algebraic ...
. It is used in almost any argument that involves a non-algebraic construction of a holomorphic function.


Uniform limits

For example, suppose that ''f''1, ''f''2, ... 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 = \oint_C \lim_ f_n(z)\,dz =\lim_ \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 In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
, the set of all bounded, analytic functions is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
with respect to the
supremum norm In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when t ...
.


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 \zeta(s) = \sum_^\infty \frac or the
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
\Gamma(\alpha) = \int_0^\infty x^ e^\,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^ e^ \, dx \,d\alpha and then using Fubini's theorem to justify changing the order of integration, getting \int_0^\infty \oint_C x^ e^ \,d\alpha \,dx = \int_0^\infty e^ \oint_C x^ \, d\alpha \,dx. Then one uses the analyticity of to conclude that \oint_C x^ \, 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 \oint_ f(z)\, dz to be zero for every closed (solid) 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. It also implies the following generalisation of the aforementioned fact about uniform limits of holomorphic functions: if ''f''1, ''f''2, ... is a sequence of holomorphic functions defined on an open set that converges to a function ''f'' uniformly on compact subsets of Ω, then ''f'' is holomorphic.


See also

*
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...
* Methods of contour integration *
Residue (complex analysis) In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for ...
* Mittag-Leffler's theorem


References

* . * . * *. * .


External links

* * {{MathWorld , urlname= MorerasTheorem , title= Morera’s Theorem Theorems in complex analysis