# Thread: Finding and classifying singularities

1. ## Finding and classifying singularities

I thought I had a grasp on this topic, but then they threw in this question:

Find and classify the singularities of
π cot(πz )
Z^2

With the use of cot I'm confused which techniques to use because I know what to use if it's sin(z) or cos(z) but not cos(z)/sin(z)

Any help insight or tricks would be greatly appreciated.

2. Hello,
I thought I had a grasp on this topic, but then they threw in this question:

Find and classify the singularities of
π cot(πz )
Z^2

With the use of cot I'm confused which techniques to use because I know what to use if it's sin(z) or cos(z) but not cos(z)/sin(z)

Any help insight or tricks would be greatly appreciated.
But you don't have to worry oO

If you're not used to the cot function, you can substitute by cos/sin.
Singularities appear at points where the function is not defined, and whether it's cot or cos/sin, they're the same.

$\displaystyle \frac{\pi \cot(\pi z)}{z^2}=\frac{\pi \cos(\pi z)}{\sin(\pi z) z^2}$
0 is obviously a singularity (not necessarily a pole because of sin(pi z))

$\displaystyle \sin(x)=0 \Leftrightarrow x=k \pi$
So here, there are also singularities for any $\displaystyle z \in \mathbb{Z}$ (the set of integers)

3. Cheers that does help, but how would I continue to classify each point?