Hey guys
Im having a panic as i should be revising, but instead im stuck on a question from my coursework, can anyone help
Find and classify the singularities of
(pi*cot(pi*x))/x^2
thanks
Hello,
cot=cos/sin
$\displaystyle \frac{\pi \cot(\pi x)}{x^2}=\frac{\pi \cos(\pi x)}{x^2 \sin(\pi x)}$
For any $\displaystyle x \in \mathbb{Z}$, there is a singularity, because thereafter, $\displaystyle \sin(\pi x)=0$
Now I believe there is a triple pole for 0, and simple pole (not too sure) for any integer in $\displaystyle \mathbb{Z}^\times$
you can check it by finding the Laurent series near these points.
The smallest value of $\displaystyle m$ such that $\displaystyle \lim_{x \rightarrow 0} x^m f(x) \neq \infty$ is $\displaystyle m = 3$ so $\displaystyle x = 0$ is a pole of order 3.
The smallest value of $\displaystyle m$ such that $\displaystyle \lim_{x \rightarrow n} (x-n)^m f(x) \neq \infty$ is $\displaystyle m = 1$ so $\displaystyle x = n$ is a simple pole (the more PC expression is pole of order 1).