Meromorphic function
Meromorphy is a property of certain complex-valued functions that are dealt with in function theory (a branch of mathematics ).
The concept of the holomorphic function is too specific for many questions in function theory . This is due to the fact that the reciprocal value of a holomorphic function has a definition gap at a zero point of and is therefore not complex differentiable there. The more general concept of the meromorphic function is therefore introduced , which can also have isolated poles .
Meromorphic functions can be represented locally as Laurent series with a terminating main part. Is an area of , the amount of on forms meromorphic functions a body .
definition
On the complex numbers
Let it be a non-empty open subset of the set of complex numbers and a further subset of that consists only of isolated points. A function is called meromorphic if it is defined for values from and holomorphic and has poles for values from . is called the set of poles of .
On a Riemannian surface
Let be a Riemann surface and an open subset of . By a meromorphic function on we understand a holomorphic function , where is an open subset, so that the following properties apply:
- The set has only isolated points.
- The following applies to every point
- .
The points from the set are called poles of . The set of all meromorphic functions on is denoted by and, if connected, forms a field . This definition is of course equivalent to the definition on the complex numbers if is a subset of them.
Examples
- All holomorphic functions are also meromorphic, since their pole set is empty.
- The reciprocal function is meromorphic; their pole set is . All rational functions are more general
- meromorphic. The set of poles is here a subset of the set of zeros of the denominator polynomial.
- For every meromorphic function , its reciprocal is also meromorphic.
- The function is not completely (and not on any neighborhood of ) meromorphic, since there is no pole but an essential singularity of this function.
- Other examples are: elliptical functions , gamma functions , Hurwitz zeta functions , modular forms , Riemann ζ functions , special functions .
Key phrases on meromorphic functions are: Mittag-Leffler's theorem , residue theorem , Riemann-Roch theorem .
literature
- E. Freitag & R. Busam - Function theory 1 , Springer-Verlag, 4th edition, ISBN 3-540-67641-4
- Otto Forster - Riemannsche surfaces , Springer-Verlag, 1977, ISBN 0-387-08034-1
- EM Chirka: Meromorphic function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).