Meromorphic function: Difference between revisions

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In the mathematical field of [[complex analysis]], a '''meromorphic function''' on an [[open set|open subset]] ''D'' of the [[complex plane]] is a [[function (mathematics)|function]] that is [[holomorphic function|holomorphic]] on all of ''D'' ''except'' for a set of [[isolated point]]s, which are [[pole (complex analysis)|pole]]s of the function.<ref name=Hazewinkel_2001>{{cite encyclopedia |editor=Hazewinkel, Michiel |year=2001 |orig-year=1994 |article=Meromorphic function |chapter-url=https://www.encyclopediaofmath.org/index.php?title=p/m063460 |encyclopedia=Encyclopedia of Mathematics |title-link=Encyclopedia of Mathematics |publisher=Springer Science+Business Media B.V. ; Kluwer Academic Publishers |ISBN=978-1-55608-010-4}} <!-- {{springer|title=Meromorphic function|id=p/m063460}} --></ref> ThisThe terminologyterm comes from the [[Ancient Greek]] ''meros'' ([[wikt:μέρος|μέρος]]), meaning "part".{{efn|Greek ''meros'' ([[wikt:μέρος|μέρος]]) means "part", in contrast with the more commonly used ''holos'' ([[wikt:ὅλος|ὅλος]]), meaning "whole".}}

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

[[File:Gamma abs 3D.png|thumb|right|The [[gamma function]] is meromorphic in the whole complex plane.]]

==Heuristic description==

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 (mathematics)#Multiplicity of a root of a polynomial|multiplicity]] of these zeros.

From an algebraic point of view, if the functions''D'' ___domain is [[connected set|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 number]]s and the [[integer]]s.

==History==

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: <math>f(z) = \csc z = \frac{1}{\sin z}.</math>

By using [[analytic continuation]] to eliminate [[removable singularity|removable singularities]], meromorphic functions can be added, subtracted, multiplied, and the quotient <math>f/g</math> can be formed unless <math>g(z) = 0</math> on a [[connected space|connected component]] of ''D''. Thus, if ''D'' is connected, the meromorphic functions form a [[field (mathematics)|field]], in fact a [[field extension]] of the [[complex numbers]].

===Higher dimensions===