In the mathematical field of complex analysis, a **meromorphic function** on an open subset *D* of the complex plane is a function that is holomorphic on all of *D**except* for a set of isolated points, which are poles of the function.^{ [1] } The term comes from the Ancient Greek *meros* (μέρος), meaning "part".^{ [lower-alpha 1] }

- Heuristic description
- Prior, alternate use
- Properties
- Higher dimensions
- Examples
- On Riemann surfaces
- Footnotes
- References

Every meromorphic function on *D* can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on *D*: any pole must coincide with a zero of the denominator.

Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at *z* and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at *z*, then one must compare the multiplicity of these zeros.

From an algebraic point of view, if the function's domain 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.

Both the field of study wherein the term is used and the precise meaning of the term changed in the 20th century. In the 1930s, in group theory, a *meromorphic function* (or *meromorph*) was a function from a group *G* into itself that preserved the product on the group. The image of this function was called an *automorphism* of *G*.^{ [2] } Similarly, a *homomorphic function* (or *homomorph*) was a function between groups that preserved the product, while a *homomorphism* was the image of a homomorph. This form of the term is now obsolete, and the related term *meromorph* is no longer used in group theory. The term * endomorphism * is now used for the function itself, with no special name given to the image of the function.

A meromorphic function is not necessarily an endomorphism, since the complex points at its poles are not in its domain, but may be in its range.

Since the poles of a meromorphic function are isolated, there are at most countably many.^{ [3] } The set of poles can be infinite, as exemplified by the function

By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient can be formed unless 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 several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, 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 a 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 ).

Unlike in dimension one, in higher dimensions there do exist compact complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori.

- All rational functions,
^{ [3] }for exampleare meromorphic on the whole complex plane. - The functions as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane.
^{ [3] } - The function 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 .
- The complex logarithm function is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane while only excluding a set of isolated points.
^{ [3] } - The function is not meromorphic in the whole plane, since the point is an accumulation point of poles and is thus not an isolated singularity.
^{ [3] } - The function is not meromorphic either, as it has an essential singularity at 0.

On a Riemann surface, every point admits an open neighborhood which is biholomorphic to an open subset of the complex plane. Thereby the notion of a meromorphic function can be defined for every Riemann surface.

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.)

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 ∞.

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.

**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 geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

In mathematics, a **holomorphic function** is a complex-valued function of one or more complex variables that is, at every point of its sub domain, complex differentiable in a neighborhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (*analytic*). Holomorphic functions are the central objects of study in complex analysis.

In complex analysis, a branch of mathematics, the **Casorati–Weierstrass theorem** describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem.

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. Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

In complex analysis, a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.

In complex analysis, a **removable singularity** of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.

In mathematics, particularly in complex analysis, a **Riemann surface** is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

In mathematics, the **complex plane** is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called **real axis**, is formed by the real numbers, and the y-axis, called **imaginary axis**, is formed by the imaginary numbers.

The **Riemann–Roch theorem** is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus *g*, in a way that can be carried over into purely algebraic settings.

In mathematics, an **affine algebraic plane curve** is the zero set of a polynomial in two variables. A **projective algebraic plane curve** is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation *h*(*x*, *y*, *t*) = 0 can be restricted to the affine algebraic plane curve of equation *h*(*x*, *y*, 1) = 0. These two operations are each inverse to the other; therefore, the phrase **algebraic plane curve** is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In mathematics, a **modular form** is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

In complex analysis, **Liouville's theorem**, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant. Equivalently, non-constant holomorphic functions on have unbounded images.

In complex analysis, a branch of mathematics, an **isolated singularity** is one that has no other singularities close to it. In other words, a complex number *z _{0}* is an isolated singularity of a function

In mathematics, a **rational function** is any function that can be defined by a **rational fraction**, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field *K*. In this case, one speaks of a rational function and a rational fraction *over K*. The values of the variables may be taken in any field *L* containing *K*. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is *L*.

In the mathematical field of complex analysis, **Nevanlinna theory** is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called it "one of the few great mathematical events of century." The theory describes the asymptotic distribution of solutions of the equation *f*(*z*) = *a*, as *a* varies. A fundamental tool is the Nevanlinna characteristic *T*(*r*, *f*) which measures the rate of growth of a meromorphic function.

In the mathematical field of complex analysis, a **branch point** of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.

In mathematics, the **Schwarzian derivative**, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.

In algebraic geometry, **divisors** are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors. Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields.

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 factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. It is named after Gösta Mittag-Leffler.

In mathematics, the **Riemann sphere**, named after Bernhard Riemann, is a model of the **extended complex plane**, the complex plane plus a point at infinity. This extended plane represents the **extended complex numbers**, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.

- ↑ Hazewinkel, Michiel, ed. (2001) [1994]. "Meromorphic function".
*Encyclopedia of Mathematics*. Springer Science+Business Media B.V. ; Kluwer Academic Publishers. ISBN 978-1-55608-010-4. - ↑ Zassenhaus, Hans (1937).
*Lehrbuch der Gruppentheorie*(1st ed.). Leipzig; Berlin: B. G. Teubner Verlag. pp. 29, 41. - 1 2 3 4 5 Lang, Serge (1999).
*Complex analysis*(4th ed.). Berlin; New York: Springer-Verlag. ISBN 978-0-387-98592-3.

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