View source for Meromorphic function

''of one complex variable in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634601.png" /> (or on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634602.png" />)''

A holomorphic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634603.png" /> which has at every singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634604.png" /> a pole (cf. [[Pole (of a function)|Pole (of a function)]], i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634605.png" /> is an isolated point of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634606.png" />, which has no limit points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634607.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634608.png" />). The collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m0634609.png" /> of all meromorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346010.png" /> is a field with respect to the usual [[pointwise operation]]s followed by redefinition at the removable singularities.

The quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346011.png" /> of two arbitrary holomorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346013.png" />, is a meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346014.png" />. Conversely, every meromorphic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346015.png" /> (or on a non-compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346016.png" />) can be expressed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346019.png" /> are holomorphic and have no common zeros in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346020.png" />. It follows that on a non-compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346021.png" /> the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346022.png" /> coincides with the field of fractions of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346023.png" /> of holomorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346024.png" />.

Every meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346025.png" /> defines a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346026.png" /> of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346027.png" /> into the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346028.png" />, which is a [[Holomorphic mapping|holomorphic mapping]] relative to the standard complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346029.png" />. Conversely, every holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346031.png" />, defines a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346033.png" />: The set of poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346034.png" /> coincides with the discrete set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346036.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346037.png" />. Thus, the meromorphic functions of one variable may be identified with the holomorphic mappings (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346038.png" />) into the Riemann sphere.

The basic problems in the theory of meromorphic functions are those concerning the existence (and construction) of meromorphic functions with prescribed singularities.

I) One is given a (closed) discrete subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346039.png" /> and, at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346040.png" />, the principal part of a Laurent expansion (cf. [[Laurent series|Laurent series]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346041.png" /></td> </tr></table>

it is required to find a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346042.png" /> with these principal parts, i.e. a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346045.png" /> is holomorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346046.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346047.png" />. If the number of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346048.png" /> is finite, then (in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346049.png" />) the problem is trivially solved by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346050.png" />. In the general case this problem is solved by the Mittag-Leffler theorem: On every non-compact Riemann surface there exists a meromorphic function with given principal parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346052.png" />. On a compact Riemann surface (for instance, a torus) this problem has in general no solution — supplementary conditions concerning the compatibility of the principal parts must be imposed.

The second basic problem is conveniently formulated in the language of divisors (cf. [[Divisor|Divisor]]), i.e. of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346053.png" /> such that for every compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346054.png" /> the number of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346055.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346056.png" /> is finite (the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346057.png" /> is called the multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346058.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346059.png" />). Divisors can explicitly be written as formal sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346061.png" /> are the points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346062.png" />; in the case of finitely many terms the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346063.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346064.png" />) is called the degree of the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346065.png" />. For a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346066.png" /> its divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346067.png" /> is equal to zero everywhere apart from the zeros and poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346068.png" />, at which the multiplicity is set equal to the order of the zero or of the pole (poles have negative orders).

II) At the points of a (closed) discrete subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346069.png" /> one is given "multiplicities" — integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346070.png" />. It is required to find a meromorphic function with zeros and poles of the respective multiplicities, i.e. a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346071.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346072.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346073.png" /> is holomorphic and does not vanish in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346075.png" />. In the case of finitely many points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346076.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346077.png" />) such a function is, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346078.png" />. In the general case the problem is solved by Weierstrass' theorem: On a non-compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346079.png" />, for every given divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346080.png" /> there is a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346081.png" /> with divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346082.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346083.png" />. For a compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346084.png" /> the holomorphic mapping into the Riemann sphere defined by a non-constant meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346085.png" /> is a branched covering, and hence the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346086.png" /> takes every value the same number of times; in particular, the number of zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346087.png" /> equals the number of its poles (multiplicities taken into account). Therefore, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346088.png" /> is necessary in order that problem II admits a solution on a compact Riemann surface. In general, it is not sufficient; a necessary and sufficient condition for the existence of a meromorphic function with a given divisor is given by Abel's theorem (see [[#References|[2]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346089.png" /> be a divisor on a compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346090.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346091.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346092.png" /> form a finite-dimensional linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346093.png" /> (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346094.png" />); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346095.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346096.png" />.

The [[Riemann–Roch theorem|Riemann–Roch theorem]] asserts that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346097.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m06346099.png" /> are the so-called canonical divisor and, respectively, the genus of the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460100.png" />. From this relation one can obtain many existence theorems (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460101.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460102.png" />, and hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460103.png" /> contains non-constant meromorphic functions). For example, on every compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460104.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460105.png" /> there is a meromorphic function which realizes a branched covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460106.png" /> with at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460107.png" /> sheets.

An important place in the theory of meromorphic functions of one complex variable is occupied by [[Value-distribution theory|value-distribution theory]] (Nevanlinna theory), which studies the distribution of the roots of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460109.png" />, when approaching the boundary of the domain.

==Meromorphic functions of several complex variables.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460110.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460111.png" /> (or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460112.png" />-dimensional complex manifold) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460113.png" /> be a (complex-) analytic subset of codimension one (or empty). A holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460114.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460115.png" /> is called a meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460116.png" /> if for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460117.png" /> one can find an arbitrarily small neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460118.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460119.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460120.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460121.png" /> holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460122.png" /> without common non-invertible factors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460123.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460124.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460125.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460126.png" /> is called the [[Polar set|polar set]] of the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460127.png" />. Its subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460128.png" />, defined locally by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460129.png" />, is called the set of (points of) indeterminacy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460130.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460131.png" /> is an analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460132.png" /> of (complex) codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460133.png" />. At each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460134.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460135.png" /> is essentially undefined: The limiting values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460136.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460138.png" />, fill up the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460139.png" />. On the other hand, at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460140.png" /> the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460141.png" /> exists, and upon redefining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460142.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460143.png" />, one obtains a holomorphic mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460144.png" /> into the Riemann sphere. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460145.png" /> is an arbitrary (possibly empty) complex-analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460146.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460147.png" />, then every holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460148.png" /> defines a meromorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460149.png" /> that is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460150.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460151.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460152.png" /> is either an analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460153.png" /> of codimension 1 or is empty. Thus, a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460154.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460155.png" /> can be defined as a holomorphic mapping into the Riemann sphere defined in the complement of an analytic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460156.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460157.png" />.

A third, completely localized, definition of meromorphic functions (equivalent to the one given above) is stated in the language of sheaves. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460158.png" /> be the [[Sheaf|sheaf]] of germs of holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460159.png" />, and for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460160.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460161.png" /> denote the field of fractions of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460162.png" /> (the stalk of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460163.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460164.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460165.png" /> is naturally endowed with the structure of a sheaf of fields, called the sheaf of germs of meromorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460166.png" />. A meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460167.png" /> is defined as a global section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460168.png" />, i.e. a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460169.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460170.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460171.png" />. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460172.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460173.png" /> are defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460174.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460176.png" />, then one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460178.png" /> are mutually prime, i.e. they have no common non-invertible factors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460179.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460180.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460181.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460182.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460183.png" />. The value at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460184.png" /> of the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460185.png" /> thus defined is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460186.png" />.

As in the one-dimensional case, the collection of all meromorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460187.png" /> forms a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460188.png" /> with respect to the pointwise algebraic operations with a subsequent redefinition at the removable singularities.

The closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460189.png" /> of the zero set of a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460190.png" />, i.e. of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460191.png" />, is an analytic subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460192.png" /> of codimension one (or empty); the set of indeterminacy is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460193.png" />. On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460194.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460195.png" /> one can define the order (multiplicity) of the zeros (or poles) of the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460196.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460197.png" /> is a regular point of the analytic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460198.png" />, then in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460199.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460200.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460201.png" /> is connected and is given by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460204.png" /> throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460205.png" />. Hence there is a maximal integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460206.png" /> such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460207.png" /> admits a holomorphic extension to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460208.png" />; this number is called the order (of the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460209.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460210.png" />, and of the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460211.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460212.png" />) of the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460213.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460214.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460215.png" /> is locally constant on the set of regular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460216.png" />. Therefore one can attach to each meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460217.png" /> its divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460218.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460219.png" /> are the irreducible components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460220.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460221.png" /> is the multiplicity (order) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460222.png" /> at the regular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460223.png" /> that belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460224.png" /> (alternative notations: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460225.png" />, etc.). On a compact complex manifold a meromorphic function is uniquely defined by its divisor, up to a multiplicative constant.

The problems solved in the one-dimensional case by the Mittag-Leffler and Weierstrass theorems are known in the higher-dimensional case as the first (additive) and the second (multiplicative) [[Cousin problems|Cousin problems]]. Due to the complicated structure of the polar set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460226.png" />, the notion of a principal part of a meromorphic function is not defined in general, and accordingly the Cousin problems are formulated as follows.

I) Suppose that an open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460227.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460228.png" /> and in each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460229.png" /> a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460230.png" /> are given; it is required to find a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460231.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460232.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460233.png" />.

II) For a given divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460234.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460235.png" />, find a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460236.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460237.png" />.

The conditions of solvability of these problems in the higher-dimensional case are considerably more stringent than in the one-dimensional case.

The problem of representing a meromorphic function as a quotient of two holomorphic functions is called the Poincaré problem. The strong Poincaré problem is to represent a meromorphic function as a quotient of holomorphic functions the germs of which at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460238.png" /> are mutually prime in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460239.png" />. The Poincaré problem is unsolvable on a compact connected complex manifold if there are non-constant meromorphic functions on it. However, this problem is solvable in every domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460240.png" /> and, in fact, in an arbitrary domain on a [[Stein manifold|Stein manifold]] (see [[#References|[7]]]). The solvability of the strong Poincaré problem follows from that of the Cousin II problem (the converse is not true).

Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460241.png" /> are said to be algebraically dependent if there is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460242.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460243.png" /> variables with complex coefficients such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460244.png" /> in the common domain of definition of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460245.png" />. The maximal number of algebraically-independent meromorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460246.png" /> is called the transcendence degree of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460247.png" />. On a compact complex manifold this number does not exceed the (complex) dimension of the manifold (Siegel's theorem); furthermore, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460248.png" /> has a finite number of generators (see [[#References|[6]]]).

On concrete complex manifolds, meromorphic functions may have supplementary properties. For instance, in the complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460249.png" /> the set of indeterminacy of any non-constant meromorphic function is not empty. Every meromorphic function on a projective algebraic variety is rational, i.e. is expressible as a quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460250.png" /> of homogeneous polynomials in homogeneous coordinates. On algebraic varieties the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460251.png" /> is quite rich. On the other hand, there exist complex manifolds (for example, some non-algebraic tori) on which every meromorphic function is constant. Higher-dimensional generalizations of the Riemann–Roch theorem are less effective, and existence theorems for various classes of meromorphic functions can only be obtained for some classes of complex manifolds.

See also [[Weierstrass theorem|Weierstrass theorem]]; [[Mittag-Leffler theorem|Mittag-Leffler theorem]]; [[Riemann–Roch theorem|Riemann–Roch theorem]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Forster, "Lectures on Riemann surfaces" , Springer (1981) (Translated from German) {{MR|0648106}} {{ZBL|0475.30002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.K. Hayman, "Meromorphic functions" , Clarendon Press (1964) {{MR|0164038}} {{ZBL|0141.07802}} {{ZBL|0115.06203}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J. Kajiwara, E. Sakai, "Generalization of Levi–Oka's theorem concerning meromorphic functions" ''Nagoya Math. J.'' , '''29''' (1967) pp. 75–84 {{MR|243106}} {{ZBL|}} </TD></TR></table>



====Comments====
Abel's theorem on meromorphic functions states the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460252.png" /> be a compact Riemann surface. A necessary and sufficient condition for a differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460253.png" /> to be the divisor of a meromorphic function is that there is a singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460254.png" />-chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460255.png" /> (cf. [[Integration on manifolds|Integration on manifolds]]) of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460256.png" /> is the boundary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460257.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460258.png" /> for all differentials of the first kind on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063460/m063460259.png" /> (cf. [[Abelian differential|Abelian differential]] and [[Integration on manifolds|Integration on manifolds]]). Cf. also [[Jacobi variety|Jacobi variety]] for another formulation of Abel's theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Sect. 10–7 {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6 {{MR|0583745}} {{ZBL|0475.30001}} </TD></TR></table>