Here's an integral that is currently giving me grey hairs:(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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

I can do the integrals of the even terms by [itex]\int_0^{\infty} = \frac{1}{2}\int_{-\infty}^{\infty}[/itex] 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

[itex]\int_0^{\infty} \frac{1}{x^{n+1}} e^{imwx} dx [/itex]

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

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# Integral from 0 to ∞ with singularity at x=0

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