Determine the nature of the singularities

[Wishbone]

I am very confused by the wording of this question, it reads:

Determine the nature of the singularities of each of the following functions and evaluate the residues (a>0)

a) 1/(z^2 + a^2)
b) 1/ (z^2+a^2)^2


Hint. fr the point at infinity use the transfor,ation w = 1/z for |z| -> 0. For the residue transform F(z)dz into g(w)dw and look at the behavior of g(w).



What does this mean, what are they asking?

 

[Wishbone]

anyone know?

 

[TD]

a) $$\frac{1}{{z^2 + a^2 }}$$

The denominator becomes zero when $z = \pm ia$, those are both poles (zeroes of the denominator but not of the nominator) of order 1 since the power of the factors is 1. To see this, and to determine the residues, factor the denominator (complex).

Have you seen how to find the residue in those points, using a limit?

b) is similar, but the poles are of order 2 here.