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A meromorphic Function is complex analytic in all but a discrete subset of its domain, and at those singularities it must go to infinity like a Polynomial (i.e., have no Essential Singularities). An equivalent definition of a meromorphic function is a complex analytic Map to the Riemann Sphere.
See also Essential Singularity, Riemann Sphere
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 382-383, 1953.