Weierstrass-Casorati theorem
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The Weierstrass-Casorati theorem in complex analysis describes the remarkable behavior of holomorphic functions near essential singularities.
Start with an open subset U of the complex plane containing the number z0, and a holomorphic function f defined on U - {z0}. The complex number z0 is called an essential singularity if there is no natural number n such that the limit
- <math>\lim_{z \to z_0} f(z) \cdot (z - z_0)^n<math>
exists. For example, the function f(z) = exp(1/z) has an essential singularity at z0 = 0, but the function g(z) = 1/z3 does not (it has a pole at 0).
The Weierstrass-Casorati theorem states that
- if f has an essential singularity at z0, and V is any neighborhood of z0 contained in U, then f(V) is dense in C.
- Or spelled out: if ε > 0 and w is any complex number, then there exists a complex number z in U with |z - z0| < ε and |f(z) - w| < ε.
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.