# Residue at a pole of non-integer order

1. Nov 13, 2005

### emob2p

Hi,
Does anyone know a straightforward way to calculate a residue at at a pole of non-integer order. I'm trying to find the residue of $$\frac {e^{ipx}}{(p - i \kappa)^\eta}$$ at $$p = i \kappa$$ where $$\eta$$ is a positive non-integer. Thanks.

I have reason to suspect it's zero, but I'd need to see the proof.

Last edited: Nov 13, 2005
2. Nov 13, 2005

### Hurkyl

Staff Emeritus
How are you defining poles and residues for non-meromorphic functions?

3. Nov 14, 2005

### emob2p

I'm not sure what non-meromorphic functions are.

4. Nov 14, 2005

### Physics Monkey

A meromorphic function is simply a function that is almost analytic in its domain. In detail, it is analytic except at a discrete set of points, and at these points it cannot have an essential singularity. Meromorphic functions are what you usually meet in the theory of residues partly because these functions are "nice" almost everywhere so many of the theorems concerning analytic functions apply. The trouble with your function is that it isn't analytic because it has a non integer power. Non integer powers are defined using the logarithm and therefore necessarily have multiple branches and require a branch cut. In this case, the cut has to start at $$p = i \kappa$$ so you have to integrate through the branch cut no matter how you define it.

Last edited: Nov 14, 2005
5. Nov 14, 2005

### emob2p

Monkey, you're everywhere. Thanks again, it makes good sense.