# Finding and classifying singularities

• Dec 22nd 2008, 05:37 PM
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.
• Dec 23rd 2008, 12:26 AM
Moo
Hello,
Quote:

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.

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

$\sin(x)=0 \Leftrightarrow x=k \pi$
So here, there are also singularities for any $z \in \mathbb{Z}$ (the set of integers)
• Dec 23rd 2008, 10:59 AM