- #1
RedSonja
- 21
- 0
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!
\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!
