In the mathematical field ofcomplex analysis, ameromorphic function on anopen subsetD of thecomplex plane is afunction that isholomorphic on all ofDexcept for a set ofisolated points, which arepoles of the function.[1] The term comes from theGreekmeros (μέρος), meaning "part".[a]
Every meromorphic function onD can be expressed as the ratio between twoholomorphic functions (with the denominator not constant 0) defined onD: 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 atz and the numerator does not, then the value of the function will approach infinity; if both parts have a zero atz, then one must compare themultiplicity of these zeros.
From an algebraic point of view, if the function's domain isconnected, then the set of meromorphic functions is thefield of fractions of theintegral domain of the set of holomorphic functions. This is analogous to the relationship between therational numbers and theintegers.
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, ingroup theory, ameromorphic function (ormeromorph) was a function from a groupG into itself that preserved the product on the group. The image of this function was called anautomorphism ofG.[2] Similarly, ahomomorphic function (orhomomorph) was a function between groups that preserved the product, while ahomomorphism was the image of a homomorph. This form of the term is now obsolete, and the related termmeromorph is no longer used in group theory.The termendomorphism 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 poles are isolated, there are at mostcountably many for a meromorphic function.[3] The set of poles can be infinite, as exemplified by the function
By usinganalytic continuation to eliminateremovable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient can be formed unless on aconnected component ofD. Thus, ifD is connected, the meromorphic functions form afield, in fact afield extension of thecomplex numbers.
Inseveral 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 theRiemann sphere: There is a set of "indeterminacy" ofcodimension two (in the given example this set consists of the origin).
Unlike in dimension one, in higher dimensions there do exist compactcomplex manifolds on which there are no non-constant meromorphic functions, for example, mostcomplex tori.
On aRiemann surface, every point admits an open neighborhoodwhich isbiholomorphic to an open subset of the complex plane. Thereby the notion of a meromorphic function can be defined for every Riemann surface.
WhenD is the entireRiemann 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-calledGAGA principle.)
For everyRiemann surface, a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not the constant function equal to ∞. The poles correspond to those complex numbers which are mapped to ∞.
On a non-compactRiemann 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.
