# coefficients of Laurent series

Suppose that $f$ is analytic in the annulus$\{z\in\mathbb{C}\,\vdots\,\,R_{1}<|z-a|,  where $R_{1}$ may be 0 and $R_{2}$ may be $\infty$.  Then the coefficients of the Laurent series  (http://planetmath.org/LaurentSeries)

 $\sum_{n=-\infty}^{\infty}c_{n}(z-a)^{n}$

of $f$ can be obtained from

 $\displaystyle c_{n}\;=\;\frac{1}{2\pi i}\oint_{\gamma}\frac{f(t)}{(t-a)^{n+1}}% \,dt\quad(n=0,\,\pm 1,\,\pm 2,\,\ldots),$ (1)

where the path (http://planetmath.org/ContourIntegral) $\gamma$ goes anticlockwise once around the point  $z=a$  within the annulus.  Especially, the residue  of $f$ in the point $a$ is

 $\displaystyle c_{-1}\;=\;\frac{1}{2\pi i}\oint_{\gamma}f(t)\,dt.$ (2)

Remark.  Usually, the Laurent series of a function  , i.e. the coefficients $c_{n}$, are not determined by using the integral formula (1), but directly from known series .  Often it is sufficient to know the value of $c_{-1}$ or the residue, which is used to compute integrals (see the Cauchy residue theorem —  cf. (2)).  There is also the usable

Rule.  In the case that the limit   $\displaystyle\lim_{z\to a}(z-a)f(z)$  exists and has a non-zero value $r$, the point  $z=a$  is a pole of the 1 for the function $f$ and

 $\operatorname{Res}(f;\,a)\;=\;r.$

Examples

1. 1.

Let  $f(z):=\frac{1}{\sin{z}}$,  and  $a=0$.  Using the Taylor series  of the complex sine we obtain

 $\lim_{z\to 0}z\frac{1}{\sin{z}}\;=\;\lim_{z\to 0}\frac{1}{1-\frac{z^{2}}{3!}+-% \ldots}\;=\;1,$

whence  $\operatorname{Res}(\frac{1}{\sin{z}};\,0)=1$.  Thus we can write

 $\oint_{\gamma}\frac{dz}{\sin{z}}\;=\;2\pi i,$

where the must be chosen such that it encloses only the pole $0$ of $\frac{1}{\sin{z}}$.

2. 2.

The Taylor series of the complex exponential function gives the Laurent series

 $e^{\frac{1}{z}}\;\equiv\;1+\frac{1}{z}+\frac{1}{2!z^{2}}+\frac{1}{3!z^{3}}+\ldots$

which shows that  $\operatorname{Res}(e^{\frac{1}{z}};\,0)=1.$

Title coefficients of Laurent series CoefficientsOfLaurentSeries 2013-03-22 15:19:22 2013-03-22 15:19:22 pahio (2872) pahio (2872) 15 pahio (2872) Result msc 30B10 LaurentSeries TechniqueForComputingResidues UniquenessOfLaurentExpansion