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 of the form
where and are entire functions with (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 on the punctured disk , where is the open unit disk.
The word derives from the Greek (meros), meaning "part," and (morphe), meaning "form" or "appearance."