Meromorphic function

• Mathematicsportal

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.

The terms holomorphic and meromorphic were introduced in 1875 by Charles Briot and Jean-Claude Bouquet, two of Augustin-Louis Cauchy's students. Holomorphic derives from the Greek ὅλος (hólos) meaning "whole", and μορφή (morphḗ) meaning "form" or "appearance" or "type", while meromorphic derives from μέρος (méros) meaning "part". A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function, resembles a rational fraction ("part") of entire functions in a domain of the complex plane.

In the 1930s, in group theory, the term meromorphic function (or meromorph) was used in a different sense: it meant a function from a group into itself that preserved the group operation. This language is obsolete, and the term endomorphism is now used for such a function.

Since poles are isolated, there are at most countably many for a meromorphic function. The set of poles can be infinite, as exemplified by the function 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 several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, f(z)=z3−2z+10z5+3z−1, are meromorphic on the whole complex plane. Furthermore, they are the only meromorphic functions on the extended complex plane.