Complex Variables: Questions on Singularities, Residues & Cauchy's P.V.

[Hypnotoad]

I have several questions on complex variables, so I will just put them all in here.

1. What are the positions and natures of the singularities and the residues at the singularities of the following functions in the z-plane, excluding the point at infinity?

a)$$f(z)=\frac{cot(\pi*z)}{(z-1)^2}$$

b)$$f(z)=\frac{1}{z(e^z-1)}$$

For part a, there is a second order singularity at z=1 and first order singularites at z=0, +/-1, +/-2, +/-3, etc... and for part b there is a first order singularity at z=0. My question on this problem is how do I find the residues? I don't have any idea how to find the residue for an infinite number of singularities and I'm also not sure how to find the exponential one.

2. Develop the first three nonzero terms of the Laurent expansion about the origin of $$f(z)=(e^z-1)^{-1}$$

I know that the expansion is given by $$f(z)=\Sum a_n(z-z_0)^n$$ with $$a_n=\frac{1}{2\pi*i}\int\frac{f(z')dz'}{(z'-z_0)^{n+1}}$$

Is there a way to easily find out which terms are the first three? Since the sum goes from negative infinity to infinty, I'm not sure what integrals to take.

3. I really don't understand Cauchy's principal value. Can anyone give an easy to understand explanation of this?

I think that is all for now. Thanks for the help.

 

[Tide]

The residue is always provided by the coefficient of the 1/(z-z_0) term so you'll need to do a taylor expansion in the case of the "second degree singularity."

 

[Hypnotoad]

The residue is always provided by the coefficient of the 1/(z-z_0) term so you'll need to do a taylor expansion in the case of the "second degree singularity."

Of the whole function, or just the term with the second order singularity in it?