coefficients of Laurent series
Suppose that is analytic in the annulus , where may be 0 and may be . Then the coefficients of the Laurent series (http://planetmath.org/LaurentSeries)
of can be obtained from
(1) |
where the path (http://planetmath.org/ContourIntegral) goes anticlockwise once around the point within the annulus. Especially, the residue of in the point is
(2) |
Remark. Usually, the Laurent series of a function, i.e. the coefficients , are not determined by using the integral formula (1), but directly from known series . Often it is sufficient to know the value of 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 exists and has a non-zero value , the point is a pole of the 1 for the function and
Examples
-
1.
Let , and . Using the Taylor series of the complex sine we obtain
whence . Thus we can write
where the must be chosen such that it encloses only the pole of .
- 2.
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 |