I've been trying to develop this excercise but can not find a way to justify the fact that the residue is -1/2
Could anyone help me please?
I've been trying to develop this excercise but can not find a way to justify the fact that the residue is -1/2
Could anyone help me please?
I like this formula for the residue of a pole of order $n$ at $z=z_0$:
$ \text{Res}[f,z_0]=\frac{1}{(n-1)!}\lim_{z\rightarrow z_0}\frac{d^{n-1}}{dz^{n-1}}((z-z_0)^nf(z)) $
Here, your poles are both order 1 (simple poles), so we have:
$ \text{Res}[f,0]=\lim_{z\rightarrow 0}zf(z)=\lim_{z\rightarrow 0}\frac{z+1}{z-2}=-\frac{1}{2} $
Note of course that this method is only really practical for low-order poles, and doesn't apply at all to essential singularities such as $\exp(1/z)$ at $z=0$.