Meromorphic function

For many problems of function theory, the notion of holomorphic function is too specific. This is because the inverse of a holomorphic function at a zero of a definition gap, and thus that there is not holomorphic. One therefore introduces the general concept of the meromorphic function, which may also have isolated poles.

Meromorphic functions can be represented locally as a Laurent series with abbrechendem main part. If U is an area of ​​, the set of meromorphic functions on U forms a body.

Definition

In the field of complex numbers

It is a non-empty subset of the open set of complex numbers, and a further portion of which consists solely of isolated points. A meromorphic function is called, if it is defined and holomorphic for values ​​and has for values ​​of poles. is called a pole- by.

On a Riemannian surface

Let be a Riemann surface and an open subset of. By a meromorphic function we mean a holomorphic function, which is an open subset, such that the following properties apply:

• The amount has only isolated points.
• For every point

The points of the set are called poles of. The set of all functions on meromorphic is designated and forms, if is connected, a body. This definition is of course equivalent to the definition of the complex numbers, if a subset of those.

Examples

• All holomorphic functions are meromorphic, as their pole set is always empty.

• The reciprocal function is meromorphic; their pole- is. More generally, all rational functions

• For any meromorphic function its reciprocal is also meromorphic.

• The tangent or cotangent function is the meromorphic.

• The function is not meromorphic at all (and no one around ), there is not a pole, but an essential singularity of this function.

• Other examples are: Elliptic functions, Gamma function, Hurwitz zeta - function, modular forms, Riemann ζ - function, special functions

Important theorems on meromorphic functions are: set of Mittag-Leffler, residue theorem, theorem of Riemann -Roch.