1
$\begingroup$

The main part of the Laurent expansion with regard to the singularity z=$-\pi$ is to be given by the following:

$$\frac{cos(z)}{(z^{2} - \pi^{2})(z+ \pi ) }$$

It is clear to me that it is a pole of the 2nd degree and the main part therefore has only 2 members. My idea was first to do a Taylor expansion of cos(z) and then to create a kind of equation system, but I was unsuccessful.

Do you have an idea how to solve the whole thing?

$\endgroup$

1 Answer 1

1
$\begingroup$

First, find the partial fraction expansion of the rational function $$\frac{1}{(z-\pi)(z+\pi)^2}$$, i.e. find $a,b,c$ such that $$\frac{1}{(z-\pi)(z+\pi)^2}=\frac{a}{z-\pi}+\frac{b}{z+\pi}+\frac{c}{(z+\pi)^2}$$. Multiply through by $\cos z$. Do each of these three series by substituting $\zeta=z+\pi$ .

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .