find the laurent series centered at $z=0$ for the rational functions below. Determine the largest open set in $C$ for which each series converges
$$\frac{1}{(z^2-1)(z^2-4)}.$$
I have no idea how to do this.
find the laurent series centered at $z=0$ for the rational functions below. Determine the largest open set in $C$ for which each series converges
$$\frac{1}{(z^2-1)(z^2-4)}.$$
I have no idea how to do this.
$$\frac{1}{(z-1)(z+1)(z-2)(z+2)}=\frac{A}{z-1}+\frac{B}{z+1}+\frac{C}{z-2}+\frac{D}{z+2}\Longrightarrow$$
$$1=A(z+1)(z^2-4)+B(z-1)(z^2-4)+C(z+2)(z^2-1)+D(z-2)(z^2+1)$$
The last equality above is one between polynomials so it must be true for any values of $\,z\,$, so for example:
$$z=-2\Longrightarrow 1=-20D\Longrightarrow D=-\frac{1}{20}$$
$$z=2\Longrightarrow 1=12C\Longrightarrow C=\frac{1}{12}$$
and etc. Thus, you can develop around $\,z=0\,$ as, for instance:
$$\frac{2}{z+2}=\frac{1}{1+\frac{z}{2}}=1-\frac{z}{2}+\frac{z^2}{4\cdot2!}-\frac{z^3}{8\cdot 3!}+\ldots$$
The above is valid whenener $\displaystyle{\left|\frac{z}{2}\right|<1\Longleftrightarrow |z|<2}\,$ , and etc.
At the end "glue" everything together putting attention on the values of $\,z\,$ for which your development are valid.