Residue of function at point
Definition
Suppose is an open subset and be a holomorphic function. Suppose is a point in such that there exists an open neighborhood such that (in other words, is an isolated singularity of ). The residue of at is defined in the following equivalent ways:
- It is the coefficient of in the Laurent series expansion of about
- It is given by the following formula, where is a small counter-clockwise circular loop about that lies completely inside :
If the following limit is a finite complex number, then that complex number equals the residue at :
(the limit is zero iff the function is holomorphic at , and is finite nonzero iff it has a simple pole at ).