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[parent] coefficients of Laurent series (Result)

Suppose that $ f$ is analytic in the annulus $ \{z\in\mathbb{C}\mid\,\, R_1 < \vert z-a\vert < R_2 \}$, where $ R_1$ may be 0 and $ R_2$ may be $ \infty$. Then the coefficients of the Laurent series expansion

$\displaystyle \sum_{n = -\infty}^\infty c_n (z-a)^n$
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 $ \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 expansions. 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 order 1 for the function $ f$ and

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

Examples

  1. Let $ f(z) := \frac{1}{\sin{z}}$, and $ a = 0$. Using the Taylor series of the complex sine we obtain
    $\displaystyle \lim_{z\to 0}z\frac{1}{\sin{z}} = \lim_{z\to 0}\frac{1}{1-\frac{z^2}{3!}+-\ldots} = 1,$
    whence $ \operatorname{Res}(\frac{1}{\sin{z}};\,0) = 1$. Thus we can write
    $\displaystyle \oint_{\gamma}\frac{dz}{\sin{z}} = 2\pi i,$
    where the path must be chosen such that it encloses only the pole 0 of $ \frac{1}{\sin{z}}$.
  2. The Taylor series of the complex exponential function gives the Laurent series
    $\displaystyle e^{\frac{1}{z}} \equiv 1+\frac{1}{z}+\frac{1}{2!z^2}+\frac{1}{3!z^3}+\ldots$
    which shows that $ \operatorname{Res}(e^{\frac{1}{z}};\,0) = 1.$



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See Also: Laurent series, technique for computing residues


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Cross-references: complex exponential function, complex sine, Taylor series, pole, limit, Cauchy residue theorem, sufficient, series, integral, function, Laurent series, residue, point, coefficients, annulus, analytic
There are 5 references to this entry.

This is version 11 of coefficients of Laurent series, born on 2005-05-30, modified 2006-10-21.
Object id is 7130, canonical name is CoefficientsOfLaurentSeries.
Accessed 3748 times total.

Classification:
AMS MSC30B10 (Functions of a complex variable :: Series expansions :: Power series )

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