technique for computing residues

The following two facts are quite useful for computing residues:

If $f$ has a pole of order at most $n+1$ at $x$, then

$$\mathrm{Res}(f;x)=\underset{y\to x}{lim}\frac{1}{n!}\frac{d^n}{d{y}^n}\left({(y-x)}^{n+1}f(y)\right).$$

If $g$ is regular at $x$ and $f$ has a simple pole at $x$, then $\mathrm{Res}(fg;x)=g(x)\mathrm{Res}(f;x)$.

Title	technique for computing residues
Canonical name	TechniqueForComputingResidues
Date of creation	2013-03-22 16:20:11
Last modified on	2013-03-22 16:20:11
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 30D30
Related topic	CoefficientsOfLaurentSeries
Related topic	ResiduesOfTangentAndCotangent