Integral solution using modified Bessel function of order 1

Question

How to solve this integral making use of Modified Bessel function of order one?

$\frac{q\gamma b}{2\pi}\int_{-\infty}^{\infty}(\gamma ^{2}\nu ^{2}t^{2}+b^{2})^{-3/2}\exp (i\omega t)dt$

Context: I was determining the power spectrum of pulse of virtual radiation for which I had: $E_{x}=-\frac{q\nu \gamma t}{(\gamma ^{2}\nu ^{2}t ^{2}+b ^{2})^{3/2}}, E_{y}=\frac{q \gamma b}{(\gamma ^{2}\nu ^{2}t ^{2}+b ^{2})^{3/2}},E_{z}=0$, $B_{x}=0,B_{y}=0,B_{z}=\beta E_{y}$

I had to use the fourier transform:

$\hat{E}(\omega )=\frac{1}{2\pi }\int E_{y}(t)\exp (i\omega t)dt$ from which I get the above Integral.

Answer 1

Hints:

• Simplify your life by letting $a=\gamma\nu$, followed by forcefully factoring a outside the radical sign.
• Further simplify your life by using the parity of the functions involved, namely $(-x)^2=x^2$, $\sin(-x)=-\sin x$, $\cos(-x)=\cos x$. $\big($Obviously, $e^{ix}=\cos x+i\sin x\big)$.
• Connect this latter simplified expression with one from this list, such as $10.32.11$, for instance.
• Notice that by using an appropriate hyperbolic substitution, similar formulas can be applied.
• If you're allowed series only, expand the integrand using the binomial series, then switch the order of summation and integration.