I am not sure how to compute the following integral:
$$ I=\int_{-\infty}^{+\infty} dx \, \frac{e^{ikx}}{(x^2 + a^2)^{\frac{3}{2}}} $$
where $a$ is a real and positive parameter.
I tried to use complex integration. However there are no simple poles due to the $3/2$ exponent. At the moment I don't know how to proceed, so if you could help me I would really appreciate it.
Thank you
Let $I(a)$ be given by the Fourier transform
$$\begin{align} I(a)&=\int_{-\infty}^\infty \frac{e^{ikx}}{(x^2+a^2)^{3/2}}\,dx\\\\ &=-\frac1a \frac{d}{da}\int_{-\infty}^\infty \frac{e^{ikx}}{(x^2+a^2)^{1/2}}\,dx\\\\ &=-\frac1a \frac{d}{da}\int_{-\infty}^\infty \frac{e^{i|ka|x}}{(x^2+1)^{1/2}}\,dx\\\\ &=-\frac1a \frac{d}{da}\int_0^\infty e^{-|ka|\cosh(x)}\,dx\tag1\\\\ &=-\frac2a \frac{d}{da}K_0(|ka|) \tag2\\\\ &=2\left|\frac{k}{a}\right|K_1(|ka|) \end{align}$$
In going from $(1)$ to $(2)$ we made use of THIS REFERENCE,