Originally Posted by

**lpd** Find the isolated singularities (if any) of the following function. For each isolated singularity, describe its nature; that is is it removeable or a pole (and of what order) or essential? In each case, calculate the residue of the function at the singularity.

a)$\displaystyle \pi cot(\pi z) - 1/z$

I got a removable singularity at z=0.

but how do I exactly work out the residue for this?

b) $\displaystyle z^{-\frac{1}{2}}$

I got none. because you cannot have a complex number which is multiplied to a power of a half. I'm not quite sure about this one...

c) $\displaystyle sin(z)sin(\frac{1}{z})$

I got a removable singularity at z=0. Not sure about this one either.... These are some hard functions that I got...

Please help me out!!