- #1

newmike

- 8

- 0

## Homework Statement

Determine the nature of the singularities of the following function and evaluate the residues.

[tex]\frac{z^{-k}}{z+1}[/tex]

for 0 < k < 1

## Homework Equations

Residue theorem, Laurent expansions, etc.

## The Attempt at a Solution

Ok this is a weird one since we've never covered anything with a non integer exponent, in fact, never with a variable exponent at all.

I realize there is a simple pole at z=-1. If I assume the numerator to be analytic and non-zero at z=-1 (which I think it true for the range of k), then I can calculate the residue at z=-1 by the simple formula: R(-1) = g(-1) / h'(-1), where g(z) is the numerator and h(z) is the denominator. If I carry through with that I get R(-1) = (-1)^(-k) which I am ok with I guess.

The problem is that I don't know what to do with the potential pole as k approaches 1. In that case we'd approach a simple pole at z=0. Alternatively, as k approached 0, we do not have a singularity at z=0 at all. So how do I handle the fact that this is a variable??

Should I try to turn this into a laurent series and go from there?

Thanks in advance.