Residue at a pole of non-integer order

[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.

 

[Hurkyl]

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

 

[emob2p]

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

 

[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.

 

[emob2p]

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