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Meromorphic functions

A meromorphic function is a complex function defined on an open subset typeset structure of the complex plane typeset structure which is holomorphic on all typeset structure except of a set of isolated points, which are poles for the function. Such functions are sometimes also called  regular functions or regular on D.

Every meromorphic function on typeset structure can be expressed as the ratio of two holomorphic functions defined on typeset structure with the denominator not identically equal 0.  Clearly, the poles then occur at the zeroes of the denominator.

The set of poles of an meromorphic function can be infinite. Take e.g. typeset structure. Since the poles of a meromorphic function are isolated, there are at most countably many  with a possible accumulation point at the complex point at infinity.

Gamma function is a function meromorphic on the whole complex plane. But the complex logarithm is not meromorphic there, because it cannot be defined on the whole complex plane except an isolated set of points.

The name comes from the Ancient Greek meros meaning part, as opposed to holos  meaning whole.

If typeset structure is also connected (to be able to use analytic continuation), then the set of meromorphic functions on typeset structure form the field of fractions of the integral domain of the set of holomorphic functions on typeset structure.  This field is actually a field extension of the field of complex numbers typeset structure.

Cite this web-page as:

Štefan Porubský: Meromorphic functions.

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