# Thread: Singularities of the function

1. ## Singularities of the function

How would you find and classify all the singularities of the function f(z) = cot (z)/z^4?

Clearly the function is undefined at 0, so has a singularity at 0, but there surely must be more to this?

2. Now you have to find the order of the pole, if it's indeed one (it seems to be)

that is to find n such that $z^nf(z)$ has a limit as z goes to 0.

3. $\lim_{z\to 0} z\cot(z) = \lim_{z\to 0 } \frac{z}{\tan(z)} =$ what ?

4. = 0 ?

so the singularity at z = 0 is a simple pole? is that correct to say?
I'm uncertain as to why it is zcot(z) and not just cot(z)/z^4?

5. Originally Posted by Roxanne123456789
= 0 ?

so the singularity at z = 0 is a simple pole? is that correct to say?
I'm uncertain as to why it is zcot(z) and not just cot(z)/z^4?
No. The previous poster is giving you a hint as to what the smallest integer value of $n$ is such that $\lim_{z \to 0} \left( z^n \frac{\cot z}{z^4}\right)$ exists and is finite.

6. Also, your problem said to "find and classify all the singularities". Since $cot(z)= \frac{cos(z)}{sin(z)}$ this function has a singularity wherever sin(z) is 0.