Weierstrass–Casorati theorem

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The Weierstrass–Casorati theorem in complex analysis describes the remarkable behavior of holomorphic functions near essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.

Start with an open subset U of the complex plane containing the number z₀, and a holomorphic function f defined on U - . The complex number z₀ is called an essential singularity if there is no natural number n such that the limit

[\lim_ f(z) \cdot (z - z_0)^n]

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.

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