Meromorphic function: Difference between revisions

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{{Short description|Class of mathematical function}}

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''poles'']]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> The term comes from the [[Greek language|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.

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* The function <math display="block"> f(z) = e^\frac{1}{z} </math> 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 <math>\mathbb{C} \setminus \{0\}</math>.

* The [[complex logarithm]] function <math display="block"> f(z) = \ln(z) </math> 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.<ref name=Lang_1999/>

* The function <math display="block"> f(z) = \csc\frac{1}{z} = \frac1{\sin\left(\frac{1}{z}\right)} </math> is not meromorphic in the whole plane, since the point <math>z = 0</math> is an [[accumulation point]] of poles and is thus not an [[isolated singularity]].<ref name=Lang_1999/>

* The function <math display="block"> f(z) = \sin \frac 1 z </math> is not meromorphic either, as it has an essential singularity at 0.

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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.

== Aporomorphy ==

In contrast to [[meromorphic functions]], which have only isolated poles, there is no universally established term in complex analysis for functions with essential singularities. While meromorphic functions are characterized by their “well-behaved” singularities, where the function diverges to infinity, functions with [[isolated singularity| isolated essential singularities]] exhibit far more complex behavior.

A function with isolated essential singularities could be called '''aporomorphic''' (from the Greek ''ἄπορος'' ''aporos, meaning “impassable” or “mysterious”''), although this term is not established in the mathematical literature. This designation would reflect the unpredictable and chaotic behavior of such functions near their singularities, as described by the Casorati–Weierstrass theorem.

== See also ==