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> This terminology comes from the [[Ancient Greek]] ''meros'' ([[wikt:μέρος|μέρος]]), meaning "part".{{efn|Greek ''meros'' ([[wikt:μέρος|μέρος]]), meaningmeans "part" is used, in technical terms contrast with the more commoncommonly 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.]]

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

From an algebraic point of view, if ''D'' 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==

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 pp|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 terminology is now obsolete. The term ''endomorphism'' is now used for the function itself, with no special name given to the image of the function. The term ''meromorph'' is no longer used in group theory.

==Properties==

Since the poles of a meromorphic function are isolated, there are at most [[countable|countably]] many.{{cn|date=October<ref 2018}}name=Lang_1999/> The set of poles can be infinite, as exemplified by the function

: <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 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===

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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

* All [[rational function]]s, for example

*::<math> f(z) = \frac{z^3 - 2z + 10}{z^5 + 3z - 1}, </math>

*:are meromorphic on the whole complex plane.

* The functions

*::<math> 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> 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> 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> f(z) = \csc\frac1zfrac{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> f(z) = \sin\frac1z </math>

*:is not meromorphic either, as it has an essential singularity at 0.

==On Riemann surfaces==

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==References==

{{reflist|25em}}

==Sources==

* {{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}}

* {{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}}

* {{springer|title=Meromorphic function|id=p/m063460}}

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