Question

How do you write the residue at z=∞ where f(z) = 1/z^3+z^5?

A)-2

B)-1

C)1

D)0

Answer 1

To calculate the residue at infinity we have the following relation :

$Res_\infty f(z) = -Res_0 g(\xi)$ with $g(\xi)=f(1/\xi)$.

Therefore, we consider the residue of $\xi^3+1/\xi^5$ at zero.

We can calculate it directly (for example, we know that the residue is the coefficient before $1/\xi$ in the Laurent series) and we find that $Res_0 g(\xi)=0$ and therefore $Res_\infty f(z)=0$ , the correct answer is D.

A faster way to obtain this result could be using the fact that the sum of residues of a meromorphic function (counting the residue at the infinity) is zero. The only residue of $f$ in $\mathbb{C}$ is the residue at zero and it is null.