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Encyclopedia > Meromorphic

A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. Every meromorphic function can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0): the poles then occur at the zeroes of the denominator.

Examples of meromorphic functions are all rational functions such as f(z) = (z3 − 2z + 1)/(z5 + 3z − 1), the functions f(z) = exp(z)/z and f(z) = sin(z)/(z − 1)2 as well as the gamma function and the Riemann zeta function. The functions f(z) = ln(z) and f(z) = exp(1/z) are not meromorphic.

By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient f/g can be formed unless g(z) = 0 for all z. Thus, the meromorphic functions form a field, in fact a field extension of the complex numbers.

In the language of Riemann surfaces, a meromorphic function is the same as a holomorphic function to the Riemann sphere which is not constant ∞. The poles correspond to those complex numbers which are mapped to ∞. Results from FactBites:

 Meromorphic function - Wikipedia, the free encyclopedia (657 words) In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: the poles then occur at the zeroes of the denominator. Meromorphic functions on an elliptic curve are also known as elliptic functions.
 PlanetMath: meromorphic extension (85 words) The meromorphic extension of an analytic function to a larger domain is unique; i.e. Occasionally, an analytic function and its meromorphic extension are denoted using the same notation. This is version 6 of meromorphic extension, born on 2006-07-29, modified 2006-10-15.
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