Meromorphic function

of one complex variable in a domain  (or on a Riemann surface )

A holomorphic function in a domain  which has at every singular point  a pole (cf. Pole (of a function), i.e.  is an isolated point of the set , which has no limit points in , and ). The collection  of all meromorphic functions in  is a field with respect to the usual pointwise operations followed by redefinition at the removable singularities.

The quotient  of two arbitrary holomorphic functions in , , is a meromorphic function in . Conversely, every meromorphic function in a domain  (or on a non-compact Riemann surface ) can be expressed as , , where  are holomorphic and have no common zeros in . It follows that on a non-compact Riemann surface  the field  coincides with the field of fractions of the ring  of holomorphic functions in .

Every meromorphic function  defines a continuous mapping  of the domain  into the Riemann sphere , which is a holomorphic mapping relative to the standard complex structure on . Conversely, every holomorphic mapping , , defines a meromorphic function  in : The set of poles of  coincides with the discrete set  and  if . Thus, the meromorphic functions of one variable may be identified with the holomorphic mappings () 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  and, at each point , the principal part of a Laurent expansion (cf. Laurent series)

it is required to find a meromorphic function  with these principal parts, i.e. a holomorphic function  in  such that  is holomorphic in a neighbourhood of  for each . If the number of points  is finite, then (in a domain ) the problem is trivially solved by the function . 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 , . 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), i.e. of mappings  such that for every compactum  the number of points  at which  is finite (the number  is called the multiplicity of  at ). Divisors can explicitly be written as formal sums , where  are the points at which ; in the case of finitely many terms the number  () is called the degree of the divisor . For a meromorphic function  its divisor  is equal to zero everywhere apart from the zeros and poles of , 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  one is given "multiplicities" — integers . It is required to find a meromorphic function with zeros and poles of the respective multiplicities, i.e. a holomorphic function  in  such that  is holomorphic and does not vanish in a neighbourhood of the point , . In the case of finitely many points  (and ) such a function is, for example, . In the general case the problem is solved by Weierstrass' theorem: On a non-compact Riemann surface , for every given divisor  there is a meromorphic function  with divisor  equal to . For a compact Riemann surface  the holomorphic mapping into the Riemann sphere defined by a non-constant meromorphic function  is a branched covering, and hence the function  takes every value the same number of times; in particular, the number of zeros of  equals the number of its poles (multiplicities taken into account). Therefore, the condition  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 [2]).

Let  be a divisor on a compact Riemann surface . The functions  satisfying the condition  form a finite-dimensional linear space  (over ); if , then .

The Riemann–Roch theorem asserts that

where  and  are the so-called canonical divisor and, respectively, the genus of the Riemann surface . From this relation one can obtain many existence theorems (if , then , and hence  contains non-constant meromorphic functions). For example, on every compact Riemann surface  of genus  there is a meromorphic function which realizes a branched covering  with at most  sheets.

An important place in the theory of meromorphic functions of one complex variable is occupied by value-distribution theory (Nevanlinna theory), which studies the distribution of the roots of the equations , , when approaching the boundary of the domain.

## Contents

* 1 Meromorphic functions of several complex variables.
* 1.1 References
* 1.2 Comments
* 1.3 References

## Meromorphic functions of several complex variables.[edit]

Let  be a domain in  (or an -dimensional complex manifold) and let  be a (complex-) analytic subset of codimension one (or empty). A holomorphic function  defined on  is called a meromorphic function in  if for every point  one can find an arbitrarily small neighbourhood  of  in  and functions  holomorphic in  without common non-invertible factors in , such that  in . The set  is called the polar set of the meromorphic function . Its subset , defined locally by the condition , is called the set of (points of) indeterminacy of ;  is an analytic subset of  of (complex) codimension . At each point  the function  is essentially undefined: The limiting values of  for , , fill up the Riemann sphere . On the other hand, at the points of  the limit  exists, and upon redefining  if , one obtains a holomorphic mapping of  into the Riemann sphere. Conversely, if  is an arbitrary (possibly empty) complex-analytic subset of  of codimension , then every holomorphic mapping  defines a meromorphic function on  that is equal to  on , where  is either an analytic subset of  of codimension 1 or is empty. Thus, a meromorphic function  in  can be defined as a holomorphic mapping into the Riemann sphere defined in the complement of an analytic subset  of codimension .

A third, completely localized, definition of meromorphic functions (equivalent to the one given above) is stated in the language of sheaves. Let  be the sheaf of germs of holomorphic functions on , and for each point  let  denote the field of fractions of the ring  (the stalk of the sheaf  over ). Then  is naturally endowed with the structure of a sheaf of fields, called the sheaf of germs of meromorphic functions in . A meromorphic function in  is defined as a global section of , i.e. a continuous mapping  such that  for all . The sets  and  are defined as follows: If , , , then one may assume that  and  are mutually prime, i.e. they have no common non-invertible factors in ; then  if , while  if . The value at a point  of the meromorphic function  thus defined is .

As in the one-dimensional case, the collection of all meromorphic functions in  forms a field  with respect to the pointwise algebraic operations with a subsequent redefinition at the removable singularities.

The closure  of the zero set of a meromorphic function , i.e. of the set , is an analytic subset of  of codimension one (or empty); the set of indeterminacy is . On  and  one can define the order (multiplicity) of the zeros (or poles) of the meromorphic function . If  is a regular point of the analytic set , then in some neighbourhood  of  the set  is connected and is given by an equation , , where  throughout . Hence there is a maximal integer  such that the function  admits a holomorphic extension to ; this number is called the order (of the zero  if , and of the pole  if ) of the meromorphic function  at the point . The function  is locally constant on the set of regular points of . Therefore one can attach to each meromorphic function in  its divisor , where  are the irreducible components of  and  is the multiplicity (order) of  at the regular points of  that belong to  (alternative notations: , 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. Due to the complicated structure of the polar set , 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  of the manifold  and in each  a meromorphic function  are given; it is required to find a meromorphic function  such that  for all .

II) For a given divisor  on , find a meromorphic function  such that .

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  are mutually prime in . 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  and, in fact, in an arbitrary domain on a Stein manifold (see [7]). The solvability of the strong Poincaré problem follows from that of the Cousin II problem (the converse is not true).

Functions  are said to be algebraically dependent if there is a polynomial  in  variables with complex coefficients such that  in the common domain of definition of the functions . The maximal number of algebraically-independent meromorphic functions on  is called the transcendence degree of the field . On a compact complex manifold this number does not exceed the (complex) dimension of the manifold (Siegel's theorem); furthermore, the field  has a finite number of generators (see [6]).

On concrete complex manifolds, meromorphic functions may have supplementary properties. For instance, in the complex projective space  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  of homogeneous polynomials in homogeneous coordinates. On algebraic varieties the field  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; Mittag-Leffler theorem; Riemann–Roch theorem.

#### References[edit]

[1] |  R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)

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[2] |  O. Forster, "Lectures on Riemann surfaces" , Springer (1981) (Translated from German) MR0648106 Zbl 0475.30002
[3] |  W.K. Hayman, "Meromorphic functions" , Clarendon Press (1964) MR0164038 Zbl 0141.07802 Zbl 0115.06203
[4] |  B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[5] |  L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001
[6] |  I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[7] |  J. Kajiwara, E. Sakai, "Generalization of Levi–Oka's theorem concerning meromorphic functions" Nagoya Math. J. , 29 (1967) pp. 75–84 MR243106

#### Comments[edit]

Abel's theorem on meromorphic functions states the following. Let  be a compact Riemann surface. A necessary and sufficient condition for a differential  to be the divisor of a meromorphic function is that there is a singular -chain  (cf. Integration on manifolds) of which  is the boundary, , such that  for all differentials of the first kind on  (cf. Abelian differential and Integration on manifolds). Cf. also Jacobi variety for another formulation of Abel's theorem.

#### References[edit]

[a1] |  G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Sect. 10–7 MR0092855 Zbl 0078.06602

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[a2] |  H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6 MR0583745 Zbl 0475.30001