removable singularity
Let be an open neighbourhood of a
point . We say that a function
has a removable singularity at
, if the complex derivative
exists for all , and
if is bounded
near .
Removable singularities can, as the name suggests, be removed.
Theorem 1
Suppose that has a removable singularity at . Then, can be holomorphically extended to all of , i.e. there exists a holomorphic such that for all .
Proof. Let be a circle centered at , oriented counterclockwise, and sufficiently small so that and its interior are contained in . For in the interior of , set
Since is a compact set, the defining limit for the derivative
converges uniformly for . Thanks to the uniform
convergence, the order of the derivative and the integral operations
can be interchanged. Hence, we may deduce that exists
for all in the interior of . Furthermore, by the Cauchy
integral formula
we have that for all , and therefore
furnishes us with the desired extension.
Title | removable singularity |
---|---|
Canonical name | RemovableSingularity |
Date of creation | 2013-03-22 12:56:01 |
Last modified on | 2013-03-22 12:56:01 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 5 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 30E99 |
Related topic | EssentialSingularity |