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The most important fact about entire functions is Liouville's theorem : an entire function which is bounded must be constant. This can be used for an elegant proof of the fundamental theorem of algebra. Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value 0.

A function that is defined on the whole complex plane except for a set of poles is called a meromorphic function.