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 [[AncientGreek 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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==Prior, alternate use==

Both the field of study wherein the term is used and the precise meaning of the term changed in the 20th&nbsp;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''.<ref>{{cite book |last=Zassenhaus |first=Hans |author-link=Hans Zassenhaus |year=1937 |title=Lehrbuch der Gruppentheorie |publisher=B. G. Teubner Verlag |___location=Leipzig; Berlin |edition=1st |pages=29, 41}}</ref> 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.

==Properties==

Since the poles of a meromorphic function are isolated, there are at most [[countable|countably]] many for a meromorphic function.<ref name=Lang_1999/> The set of poles can be infinite, as exemplified by the function <math display="block">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]].

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==Examples==

* All [[rational function]]s,<ref name=Lang_1999>{{cite book |last=Lang |first=Serge |author-link=Serge Lang |year=1999 |title=Complex analysis |publisher=[[Springer-Verlag]] |___location=Berlin; New York |edition=4th |isbn=978-0-387-98592-3}}</ref> for example <math display="block"> f(z) = \frac{z^3 - 2z + 10}{z^5 + 3z - 1}, </math> are meromorphic on the whole complex plane. Furthermore, they are the only meromorphic functions on the [[riemann sphere|extended complex plane]].

* The functions <math display="block"> f(z) = \frac{e^z}{z} \quad\text{and}\quad f(z) = \frac{\sin{z}}{(z-1)^2} </math> as well as the [[gamma function]] and the [[Riemann zeta function]] are meromorphic on the whole complex plane.<ref name=Lang_1999/>

* 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 [[Limit point|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 ==