Encyclopedia > Weierstrass-Casorati theorem

  Article Content

Weierstrass-Casorati theorem

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Anna Karenina

... Its theme is the institution of marriage and its relation to society and morality. The novel intially appeared serially in the periodical Ruskii Vestnik ("Russia ...

 
 
 
This page was created in 23.6 ms