Start with an open subset U of the complex plane containing the number z_{0}, and a holomorphic function f defined on U  {z_{0}}. The complex number z_{0} is called an essential singularity if there is no natural number n such that the limit
exists. For example, the function f(z) = exp(1/z) has an essential singularity at z_{0} = 0, but the function g(z) = 1/z^{3} does not (it has a pole at 0).
The WeierstrassCasorati theorem states that
The theorem is considerably strengthened by Picard's great theorem[?], which states, in the notation above, that f assumes every complex value, with one possible exception, infinitely often on V.
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