Meromorphic Function

A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function is a function f(z) of the form


where g(z) and h(z) are entire functions with h(z)!=0 (Krantz 1999, p. 64).

A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. A meromorphic function with an infinite number of poles is exemplified by csc(1/z) on the punctured disk U=D\{0}, where D is the open unit disk.

An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere.

The word derives from the Greek muepsilonrhoomicronsigma (meros), meaning "part," and muomicronrhophieta (morphe), meaning "form" or "appearance."

See also

Analytic Function, Entire Function, Essential Singularity, Holomorphic Function, Pole, Real Analytic Function, Riemann Sphere

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Knopp, K. "Meromorphic Functions." Ch. 2 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 34-57, 1996.Krantz, S. G. "Meromorphic Functions and Singularities at Infinity." §4.6 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 63-68, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 382-383, 1953.

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Meromorphic Function

Cite this as:

Weisstein, Eric W. "Meromorphic Function." From MathWorld--A Wolfram Web Resource.

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