Integral from 0 to ∞ with singularity at x=0

[RedSonja]

Here's an integral that is currently giving me grey hairs:

$\int_0^{\infty} \frac{1}{x} \exp(i \frac{k}{x}(a-c \cos(\theta + wx))) dx$

I've tried different approaches like contour integration around $x=0$ and replacing the exponential by its Taylor sum to have:

$\int_0^{\infty} \sum_{n=0}^{\infty} \frac{1}{x^{n+1}\;n!} (i k (a-c \cos(\theta + wx)))^n dx$

I can do the integrals of the even terms by $\int_0^{\infty} = \frac{1}{2}\int_{-\infty}^{\infty}$ and residues, but I don't know how to handle the odd terms.

Going to sum-extremes I've rewritten the integral to a form where I only need to do integrals of the type

$\int_0^{\infty} \frac{1}{x^{n+1}} e^{imwx} dx$

with m a positive or negative integer, but even here I'm stuck. Please help!

 

[rollingstein]

Maybe this is a stupid question, but is the integration over the real axis?

 

[RedSonja]

Maybe this is a stupid question, but is the integration over the real axis?

Yes, x is real. But since the integrand goes to zero for $x\rightarrow \infty$ the direction of integration in the complex plane shouldn't alter the integral...

 

[RedSonja]

Hmm.
Is it possible that the latter of the three integrals is simply the gamma-function?