coefficients of Laurent series

Suppose that f is analytic in the annulus{zR1<|z-a|<R2},  where R1 may be 0 and R2 may be .  Then the coefficients of the Laurent seriesMathworldPlanetmath (


of f can be obtained from

cn=12πiγf(t)(t-a)n+1dt(n=0,±1,±2,), (1)

where the path ( γ goes anticlockwise once around the point  z=a  within the annulus.  Especially, the residueDlmfPlanetmath of f in the point a is

c-1=12πiγf(t)𝑑t. (2)

Remark.  Usually, the Laurent series of a functionMathworldPlanetmath, i.e. the coefficients cn, 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   limza(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



  1. 1.

    Let  f(z):=1sinz,  and  a=0.  Using the Taylor seriesMathworldPlanetmath of the complex sine we obtain

    limz0z1sinz=limz011-z23!+-= 1,

    whence  Res(1sinz; 0)=1.  Thus we can write

    γdzsinz= 2πi,

    where the must be chosen such that it encloses only the pole 0 of 1sinz.

  2. 2.

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

    e1z 1+1z+12!z2+13!z3+

    which shows that  Res(e1z; 0)=1.

Title coefficients of Laurent series
Canonical name CoefficientsOfLaurentSeries
Date of creation 2013-03-22 15:19:22
Last modified on 2013-03-22 15:19:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Result
Classification msc 30B10
Related topic LaurentSeries
Related topic TechniqueForComputingResidues
Related topic UniquenessOfLaurentExpansion