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 Thisname=Hazewinkel_2001>{{cite terminologyencyclopedia |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 asmore opposedcommonly toused ''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 [[zeroMultiplicity (complex analysismathematics)#Multiplicity of a root of a polynomial|multiplicitiesmultiplicity]] of these zeros.

From an algebraic point of view, if the function''D''s ___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.

==Prior, alternate use==

==History==

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 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 terminologyform isof now obsolete. Thethe term ''endomorphism'' is now used for the function itselfobsolete, with no special name given toand the image of the function. Therelated 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>citation needed<name=Lang_1999/ref> The set of poles can be infinite, as exemplified by the function <math display="block">f(z) = \csc z = \frac{1}{\sin z}.</math>

: <math>f(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===

In [[several complex variables]], a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, <math>f(z_1, z_2) = z_1 / z_2</math> is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as a holomorphic function with values in the [[Riemann sphere]]: There is a set of "indeterminacy" of [[codimension]] two (in the given example this set consists of the origin <math>(0, 0)</math>).

Unlike in dimension one, in higher dimensions there do exist compact [[complex manifold]]s on which there are no non-constant meromorphic functions, for example, most [[complex torus|complex tori]].

==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]].

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

*: 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/>

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

*: 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>.

*:are meromorphic on the whole complex plane.

*: 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 functions

*: 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 The functionname=Lang_1999/>

*::<math> f(z) = \frac{e^z}{z} \quad\text{and}\quad f(z) = \frac{\sin{z}}{(z-1)^2} </math>

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

*:as well as the [[gamma function]] and the [[Riemann zeta function]] are meromorphic on the whole complex plane.

* 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.

* The function

*::<math> f(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. The function

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

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

==On Riemann surfaces==

On a [[Riemann surface]], every point admits an open neighborhood

which is [[homeomorphismbiholomorphism|homeomorphicbiholomorphic]] to an open subset of the complex plane. Thereby the notion of a meromorphic function can be defined for every Riemann surface.

When ''D'' is the entire [[Riemann sphere]], the field of meromorphic functions is simply the field of rational functions in one variable over the complex field, since one can prove that any meromorphic function on the sphere is rational. (This is a special case of the so-called [[GAGA]] principle.)

For every [[Riemann surface]], a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not the constant function equal to ∞. The poles correspond to those complex numbers which are mapped to ∞.

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 ==

Meromorphic functions on an [[elliptic curve]] are also known as [[elliptic function]]s.

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.

==References==

{{Reflist}}

== See also ==

* {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Complex analysis | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | edition=4th | isbn=978-0-387-98592-3 | year=1999}}

*[[Cousin problems]]

* {{Citation | last1=Zassenhaus | first1=Hans | author1-link=Hans Zassenhaus | title=Lehrbuch der Gruppentheorie | publisher=[[Verlag und Druck von B.G.Teubner]] | ___location=Leipzig, Berlin | edition=1st | year=1937}}

*[[Mittag-Leffler's theorem]]

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

*[[Weierstrass factorization theorem]]

==Footnotes==

{{notelist|1}}

==References==

{{reflist|25em}}

{{Authority control}}