TITLE: Finding poles, indicating their order and then computing their residues
QUESTION [35 upvotes]: I really don't understand the concept behind finding poles in Complex Analysis and I can't find anything on the internet or in books that helps me grasp the concept...
The following are past exam questions that I'm looking at but don't know where to go with with them in order to find their poles, indicate their order to then compute their residues.. Any help would be greatly appreciated...
$(i) f(z)=\dfrac{\sin z }{(z-1)\sinh z}$
$(ii)g(z)= \dfrac{\sin z}{z(z^2+4)}$

REPLY [7 votes]: (i) The poles of $f(z)=\dfrac{\sin z}{(z-1)\sinh z}$ will be the poles of $\sin z$ along with the zeroes of $(z-1)\sinh z$). Since $\sin(z)$ is analytic it will have no poles.
Thus the poles will by $z=1$ and the zeros of $\sinh z$, which are $z=n\pi i$ for $z\in \mathbb{Z}$. and they will all be simple.
At $z=1$, we can use the formula $Res(f,c)=\lim_{z\rightarrow c}(z-c)f(z)$. Thus $Res(f,1)=\lim_{z\rightarrow 1}\dfrac{\sin z}{\sinh z}=\dfrac{\sin 1}{\sinh 1}$ 
For the $z=n\pi i$ poles, we want to use the formula $Res(\dfrac{g}{h},c)=\dfrac{g(c)}{h^{\prime}(c)}$, so $Res(f,n\pi i)=\dfrac{sin(n\pi i)}{(n\pi i-1)\cosh (n\pi i)}=\dfrac{\sinh(n\pi)}{n\pi i-1} $