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In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. (The terminology comes from the Ancient Greek “meros” (μέρος), meaning part, as opposed to “holos” (ὅλος), meaning whole.) Such functions are sometimes said to be regular functions or regular on D. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an...
Note: This article contains special characters. ...
Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: the poles then occur at the zeroes of the denominator. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
Intuitively then, a meromorphic function is a ratio of two nice (holomorphic) functions. Such a function will still be "nice", except at the points where the denominator of the fraction is zero, when the value of the function will be infinite. Wikipedia does not have an article with this exact name. ...
Wikipedia does not have an article with this exact name. ...
The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...
From an algebraic point of view, if D is connected, then the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions. This is analogous to the relationship between  , the rational numbers, and  , the integers. In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ...
In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ...
In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
The integers are commonly denoted by the above symbol. ...
Examples
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 - are meromorphic on the whole complex plane.
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 and  - as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane.
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 - is defined in the whole complex plane except for the origin, 0. However, 0 is not a pole of this function, rather an essential singularity. Thus, this function is not meromorphic in the whole complex plane. However, it is meromorphic (even holomorphic) on C{0}.
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 - is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane except an isolated set of points.
In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...
The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...
In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...
In complex analysis, an essential singularity of a function is a severe singularity near which the function exhibits extreme behavior. ...
The natural logarithm is the logarithm to the base e, where e is equal to 2. ...
Properties Since the poles of a meromorphic function are isolated, there are at most countably many. The set of poles can be infinite, as exemplified by the function In mathematics the term countable set is used to describe the size of a set, e. ...
 By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient f / g can be formed unless g(z) = 0 on a connected component of D. Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...
In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be defined without creating any problems. ...
Connected components come up in topology and in graph theory, two related branches of mathematics. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
Meromorphic functions on Riemann surfaces On a Riemann surface every point admits an open neighborhood which is isomorphic to an open subset of the complex plane. Thereby the notion of a meromorphic function can be defined for every Riemann surface. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...
When D is the entire Riemann sphere, the field of meromorphic functions is simply the field of rational functions in one variable over the complex field, since one can prove that any meromorphic function on the sphere is rational. (This is a special case of the so-called GAGA principle.) A rendering of the Riemann Sphere. ...
Look up gaga in Wiktionary, the free dictionary. ...
For every Riemann surface, a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not constant ∞. The poles correspond to those complex numbers which are mapped to ∞. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...
On a non-compact Riemann surface every meromorphic function can be realized as a quotient of two (globally defined) holomorphic functions. In contrast, on a compact Riemann surface every holomorphic function is constant, while there always exist non-constant meromorphic functions. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...
Meromorphic functions on an elliptic curve are also known as elliptic functions. In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...
In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...
Higher dimensions In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, f(z1,z2)=z1/z2 is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as holomorphic function with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two (in the given example this set consists of the origin (0,0)). The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ...
A rendering of the Riemann Sphere. ...
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. ...
Unlike in dimension one, in higher dimensions there do exist complex manifolds on which there are no non-constant meromorphic functions. In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...
References - Serge Lang, Complex Analysis, Springer, 2003. ISBN 0-387-98592-1.
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