In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Morera's theorem, named after
Giacinto Morera
Giacinto Morera (18 July 1856 – 8 February 1909), was an Italian engineer and mathematician. He is known for Morera's theorem in the theory of functions of a complex variable and for his work in the theory of linear elasticity.
Biography
L ...
, gives an important criterion for proving that a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
.
Morera's theorem states that a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
''D'' in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
that satisfies
for every closed piecewise ''C''
1 curve
in ''D'' must be holomorphic on ''D''.
The assumption of Morera's theorem is equivalent to ''f'' locally having an
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
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 between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
; this is
Cauchy's integral theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in t ...
, stating that the
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
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 it's still 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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. Fix a point ''z''
0 in ''D'', and for any
, let
be a piecewise ''C''
1 curve such that
and
. Then define the function ''F'' to be
To see that the function is well-defined, suppose
is another piecewise ''C''
1 curve such that
and
. The curve
(i.e. the curve combining
with
in reverse) is a closed piecewise ''C''
1 curve in ''D''. Then,
And it follows that
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
In complex analysis, a complex-valued function f of a complex variable z:
*is said to be holomorphic at a point a if it is differentiable at every point within some open disk centered at a, and
* is said to be analytic at a if in some open disk ...
, i.e. can be represented by a convergent power series, 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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. 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
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 ''f'' must be holomorphic. This fact can be used to show that, for any
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
, the set of all
bounded, analytic functions is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
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 th ...
.
Infinite sums and integrals
Morera's theorem can also be used in conjunction with
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
and the
Weierstrass M-test
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous t ...
to show the analyticity of functions defined by sums or integrals, such as the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
or the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
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 to conclude that
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
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 Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differ ...
*
Methods of contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of complex analysis.
...
*
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 an ...
*
Mittag-Leffler's theorem In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass fact ...
References
* .
* .
*
*.
* .
External links
*
* {{MathWorld , urlname= MorerasTheorem , title= Morera’s Theorem
Theorems in complex analysis