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 . ${\displaystyle {\tfrac {1}{f}}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle {\tfrac {1}{f}}}$

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 . ${\displaystyle U}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle U}$

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 . ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle P_{f}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle D\setminus P_{f}}$ ${\displaystyle P_{f}}$ ${\displaystyle P_{f}}$ ${\displaystyle f}$

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: ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon Y'\rightarrow \mathbb {C} }$ ${\displaystyle Y'\subset Y}$

• The set has only isolated points. ${\displaystyle P_{f}:=Y\setminus Y'}$
• The following applies to every point ${\displaystyle p\in Y\setminus Y'}$

${\displaystyle \lim _{x\rightarrow p}|f(x)|=\infty }$.

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. ${\displaystyle Y\setminus Y'}$ ${\displaystyle f}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {M}}(Y,\mathbb {C} )}$ ${\displaystyle Y}$ ${\displaystyle X}$

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 ${\displaystyle z\mapsto {\tfrac {1}{z}}}$ ${\displaystyle \{0\}}$

  ${\displaystyle z\mapsto {\frac {a_{m}z^{m}+\dotsb +a_{0}}{b_{n}z^{n}+\dotsb +b_{0}}}}$

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. ${\displaystyle f\neq 0}$ ${\displaystyle {\tfrac {1}{f}}}$

• The tangent or cotangent function is 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. ${\displaystyle z\mapsto e^{1/z}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle 0}$ ${\displaystyle 0}$

• 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 ).