1. ## Calculating a residue

does anyone know how to work out the residue of this at 0

Res(f,0)= .5 lim (2*pi^3*z*cot(pi*z) -2*pi^2)cosec^2(pi*z)
(as z tends to 0)

But i cant work out the limit, seeing as everytime i work it out i get different answers, i was told to use taylor series though

2. Originally Posted by pols89
does anyone know how to work out the residue of this at 0

Write cot(πz) as cos(πz)/sin(πz) and cosec(πz) as 1/sin(πz). Then you get $\displaystyle \lim_{z\to0}\frac{\pi^2(\pi z\cos(\pi z) - \sin(\pi z))}{\sin^3(\pi z)}$. Now use the power series expansions of sin and cos to find the lowest power of z with nonzero coefficient. I make the limit $\displaystyle -\pi^2/3$.