Question: Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $z\cos{\frac{1}{z}}$ at $z = \infty$.
Answer:
Let $w = \frac{1}{z}$, then we are finding Laurent series for $\frac{\cos{w}}{w}$ at $w = 0$.
$\cos{w} = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}w^{2n}$,
so $z\cos{\frac{1}{z}} = \frac{\cos{w}}{w} = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}w^{2n-1} = \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}(\frac{1}{z})^{2n-1}$.
Note that $\frac{\cos{w}}{w} = \frac{1}{w}(1-\frac{w^2}{2!}+\frac{w^4}{4!}-...) = \frac{1}{w}- \frac{w}{2!}+\frac{w^3}{4!}-...$,
so the residue of the function is 1 as the coefficient for the term $\frac{1}{w}$ is 1.
$z\cos{\frac{1}{z}} = \frac{1}{\frac{1}{z}}- \frac{\frac{1}{z}}{2!}+\frac{(\frac{1}{z})^3}{4!}-... = z-\frac{1}{z(2!)}+\frac{1}{z^3(4!)}-...$
