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<title>Meromorphic function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Meromorphic function</span></span>

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<p>In the mathematical field of <a href="Complex_analysis" title="Complex analysis">complex analysis</a>, a <b>meromorphic function</b> on an <a href="Open_set" title="Open set">open subset</a> <i>D</i> of the <a href="Complex_plane" title="Complex plane">complex plane</a> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that is <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> on all of <i>D</i> <i>except</i> for a set of <a href="Isolated_point" title="Isolated point">isolated points</a>, which are <a href="Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)"><i>poles</i></a> of the function.<sup id="cite_ref-Hazewinkel_2001_1-0" class="reference"><a href="#cite_note-Hazewinkel_2001-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The term comes from the <a href="Greek_language" title="Greek language">Greek</a> <i>meros</i> (<a href="https://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%81%CE%BF%CF%82" class="extiw external" title="wikt:μέρος">μέρος</a>), meaning "part".<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup>

</p><p>Every meromorphic function on <i>D</i> can be expressed as the ratio between two <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a> (with the denominator not constant 0) defined on <i>D</i>: any pole must coincide with a zero of the denominator.

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<div class="mw-heading mw-heading2"><h2 id="Heuristic_description">Heuristic description</h2></div>

<p>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 <i>z</i> and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at <i>z</i>, then one must compare the <a href="Multiplicity_(mathematics)#Multiplicity_of_a_root_of_a_polynomial" title="Multiplicity (mathematics)">multiplicity</a> of these zeros.

</p><p>From an algebraic point of view, if the function's domain is <a href="Connected_set" class="mw-redirect" title="Connected set">connected</a>, then the set of meromorphic functions is the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of the <a href="Integral_domain" title="Integral domain">integral domain</a> of the set of holomorphic functions. This is analogous to the relationship between the <a href="Rational_number" title="Rational number">rational numbers</a> and the <a href="Integer" title="Integer">integers</a>.

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<div class="mw-heading mw-heading2"><h2 id="Prior,_alternate_use">Prior, alternate use</h2></div>

<p>Both the field of study wherein the term is used and the precise meaning of the term changed in the 20th century. In the 1930s, in <a href="Group_theory" title="Group theory">group theory</a>, a <i>meromorphic function</i> (or <i>meromorph</i>) was a function from a group <i>G</i> into itself that preserved the product on the group. The image of this function was called an <i>automorphism</i> of <i>G</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Similarly, a <i>homomorphic function</i> (or <i>homomorph</i>) was a function between groups that preserved the product, while a <i>homomorphism</i> was the image of a homomorph. This form of the term is now obsolete, and the related term <i>meromorph</i> is no longer used in group theory.

The term <i><a href="Endomorphism" title="Endomorphism">endomorphism</a></i> is now used for the function itself, with no special name given to the image of the function.

</p><p>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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>Since poles are isolated, there are at most <a href="Countable" class="mw-redirect" title="Countable">countably</a> many for a meromorphic function.<sup id="cite_ref-Lang_1999_4-0" class="reference"><a href="#cite_note-Lang_1999-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The set of poles can be infinite, as exemplified by the function <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\csc z={\frac {1}{\sin z}}.}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)=\csc z={\frac {1}{\sin z}}.}</annotation>

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</math></span></span>

</p><p>By using <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a> to eliminate <a href="Removable_singularity" title="Removable singularity">removable singularities</a>, meromorphic functions can be added, subtracted, multiplied, and the quotient <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f/g}">

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<annotation encoding="application/x-tex">{\displaystyle f/g}</annotation>

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</math></span><img src="./b5c6b7962d3532e248f07cd42b1bdc9e007b137d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.557ex; height:2.843ex;" alt="{\displaystyle f/g}" loading="lazy"></span> can be formed unless <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(z)=0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g(z)=0}</annotation>

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</math></span><img src="./be3860c571233d32e08a7879c207a04be0804d46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.274ex; height:2.843ex;" alt="{\displaystyle g(z)=0}" loading="lazy"></span> on a <a href="Connected_space" title="Connected space">connected component</a> of <i>D</i>. Thus, if <i>D</i> is connected, the meromorphic functions form a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>, in fact a <a href="Field_extension" title="Field extension">field extension</a> of the <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>.

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<div class="mw-heading mw-heading3"><h3 id="Higher_dimensions">Higher dimensions</h3></div>

<p>In <a href="Several_complex_variables" class="mw-redirect" title="Several complex variables">several complex variables</a>, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z_{1},z_{2})=z_{1}/z_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle f(z_{1},z_{2})=z_{1}/z_{2}}</annotation>

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</math></span><img src="./4f1adbb269ae07286bb7178fbfb9858af04c43b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.924ex; height:2.843ex;" alt="{\displaystyle f(z_{1},z_{2})=z_{1}/z_{2}}" loading="lazy"></span> 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 <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>: There is a set of "indeterminacy" of <a href="Codimension" title="Codimension">codimension</a> two (in the given example this set consists of the origin <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation>

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</math></span><img src="./5d630d3e781a53b0a3559ae7e5b45f9479a3141a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}" loading="lazy"></span>).

</p><p>Unlike in dimension one, in higher dimensions there do exist compact <a href="Complex_manifold" title="Complex manifold">complex manifolds</a> on which there are no non-constant meromorphic functions, for example, most <a href="Complex_torus" title="Complex torus">complex tori</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>All <a href="Rational_function" title="Rational function">rational functions</a>,<sup id="cite_ref-Lang_1999_4-1" class="reference"><a href="#cite_note-Lang_1999-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> for example <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {z^{3}-2z+10}{z^{5}+3z-1}},}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {z^{3}-2z+10}{z^{5}+3z-1}},}</annotation>

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</math></span></span> are meromorphic on the whole complex plane. Furthermore, they are the only meromorphic functions on the <a href="Riemann_sphere" title="Riemann sphere">extended complex plane</a>.</li>

<li>The functions <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {e^{z}}{z}}\quad {\text{and}}\quad f(z)={\frac {\sin {z}}{(z-1)^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {e^{z}}{z}}\quad {\text{and}}\quad f(z)={\frac {\sin {z}}{(z-1)^{2}}}}</annotation>

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</math></span></span> as well as the <a href="Gamma_function" title="Gamma function">gamma function</a> and the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> are meromorphic on the whole complex plane.<sup id="cite_ref-Lang_1999_4-2" class="reference"><a href="#cite_note-Lang_1999-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

<li>The function <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=e^{\frac {1}{z}}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f(z)=e^{\frac {1}{z}}}</annotation>

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</math></span></span> is defined in the whole complex plane except for the origin, 0. However, 0 is not a pole of this function, rather an <a href="Essential_singularity" title="Essential singularity">essential singularity</a>. Thus, this function is not meromorphic in the whole complex plane. However, it is meromorphic (even holomorphic) on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \setminus \{0\}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \setminus \{0\}}</annotation>

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</math></span><img src="./91caa89c0163b2f76279b22233ea55460f82a63b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.36ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} \setminus \{0\}}" loading="lazy"></span>.</li>

<li>The <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> function <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\ln(z)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(z)=\ln(z)}</annotation>

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</math></span></span> 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.<sup id="cite_ref-Lang_1999_4-3" class="reference"><a href="#cite_note-Lang_1999-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

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<annotation encoding="application/x-tex">{\displaystyle f(z)=\csc {\frac {1}{z}}={\frac {1}{\sin \left({\frac {1}{z}}\right)}}}</annotation>

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</math></span></span> is not meromorphic in the whole plane, since the point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}">

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<annotation encoding="application/x-tex">{\displaystyle z=0}</annotation>

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</math></span><img src="./b92bfc06485cc90286474b14a516a68d8bfdd7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}" loading="lazy"></span> is an <a href="Accumulation_point" title="Accumulation point">accumulation point</a> of poles and is thus not an <a href="Isolated_singularity" title="Isolated singularity">isolated singularity</a>.<sup id="cite_ref-Lang_1999_4-4" class="reference"><a href="#cite_note-Lang_1999-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

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<annotation encoding="application/x-tex">{\displaystyle f(z)=\sin {\frac {1}{z}}}</annotation>

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</math></span></span> is not meromorphic either, as it has an essential singularity at 0.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="On_Riemann_surfaces">On Riemann surfaces</h2></div>

<p>On a <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>, every point admits an open neighborhood

which is <a href="Biholomorphism" title="Biholomorphism">biholomorphic</a> to an open subset of the complex plane. Thereby the notion of a meromorphic function can be defined for every Riemann surface.

</p><p>When <i>D</i> is the entire <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>, 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 <a href="GAGA" class="mw-redirect" title="GAGA">GAGA</a> principle.)

</p><p>For every <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>, 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 ∞.

</p><p>On a non-compact <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>, 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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Aporomorphy">Aporomorphy</h2></div>

<p>In contrast to <a href="Meromorphic_functions" class="mw-redirect" title="Meromorphic functions">meromorphic functions</a>, 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 <a href="Isolated_singularity" title="Isolated singularity"> isolated essential singularities</a> exhibit far more complex behavior.

</p><p>A function with isolated essential singularities could be called <b>aporomorphic</b> (from the Greek <i>ἄπορος</i> <i>aporos, meaning “impassable” or “mysterious”</i>), 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.

</p>

<ul><li><a href="Cousin_problems" title="Cousin problems">Cousin problems</a></li>

<li><a href="Mittag-Leffler's_theorem" title="Mittag-Leffler's theorem">Mittag-Leffler's theorem</a></li>

<li><a href="Weierstrass_factorization_theorem" title="Weierstrass factorization theorem">Weierstrass factorization theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2></div>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Greek <i>meros</i> (<a href="https://en.wiktionary.org/wiki/%CE%BC%CE%AD%CF%81%CE%BF%CF%82" class="extiw external" title="wikt:μέρος">μέρος</a>) means "part", in contrast with the more commonly used <i>holos</i> (<a href="https://en.wiktionary.org/wiki/%E1%BD%85%CE%BB%CE%BF%CF%82" class="extiw external" title="wikt:ὅλος">ὅλος</a>), meaning "whole".</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHazewinkel,_Michiel2001" class="citation encyclopaedia cs1">Hazewinkel, Michiel, ed. (2001) [1994]. <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=p/m063460">"Meromorphic function"</a>. <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics"><i>Encyclopedia of Mathematics</i></a>. Springer Science+Business Media B.V.; Kluwer Academic Publishers. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-55608-010-4</bdi>.</cite> </span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFZassenhaus1937" class="citation book cs1"><a href="Hans_Zassenhaus" title="Hans Zassenhaus">Zassenhaus, Hans</a> (1937). <i>Lehrbuch der Gruppentheorie</i> (1st ed.). Leipzig; Berlin: B. G. Teubner Verlag. pp. 29, 41.</cite></span>

</li>

<li id="cite_note-Lang_1999-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lang_1999_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lang_1999_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lang_1999_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Lang_1999_4-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Lang_1999_4-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFLang1999" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1999). <i>Complex analysis</i> (4th ed.). Berlin; New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98592-3</bdi>.</cite></span>

</li>

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</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q217616#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1479" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q217616#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1479" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4136862-9">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Functions, Meromorphic"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85052343">United States</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00574498">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="meromorfní funkce"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph1118153&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000169913&P_CON_LNG=ENG">Latvia</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007553157505171">Israel</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="http://esu.com.ua/search_articles.php?id=67459">Encyclopedia of Modern Ukraine</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/cb6b10fe-c171-4bcf-ab26-e3f9fb24b65f">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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