Meromorphic functions

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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. (The terminology comes from the Ancient Greek “meros” (μέρος), meaning part, as opposed to “holos” (ὅλος), meaning whole.)

In particular, every holomorphic function can be considered as meromorphic.

Many of special functions are meromorphic.

For example, factorial is meromorphic in the whole complex plane (it has countable set of poles at the negative integer values of the argument), but logarithm is not, because it has cutline at the negative part of the real axis. However, the same logarithm becomes meromorphic being considered on the domain of numbers with positive real part.