I am looking for suggestions to evaluate this F.T which involves bessel function of the second kind (of zero order) I = $$\int_{-\infty}^{\infty} K_0(\sqrt{z^2+a^2}) e^{i kz} dz $$ Can this Integral I be solved analytically? Are there any analytical approximations I can use?
Thanks
$$ (1) \quad \int_{-\infty}^\infty K_0(\sqrt{z^2+a^2}) \ e^{i k z} dz = \pi \, \frac{e^{-a\sqrt{k^2+1}}}{\sqrt{k^2+1}}$$
The proof uses the Dirac delta function relation and a specialization of a formula from Gradshteyn and Ryzhik, 6.596.7. With that formula, let $\mu = 1/2$, $\nu=-1/2$, $\beta = z$, and $\alpha = a.$ Use the well-known reduction of the Bessel functions at odd half-integers. Then you'll get
$$ (2) \quad K_0(\sqrt{z^2+a^2}) = \int_0^\infty \cos{zt} \ \frac{e^{-a\sqrt{t^2+1}}}{\sqrt{t^2+1}} \ dt $$
Insert (2) into the left-hand side of (1) and interchange $\int.$ The inner integral is $$ \frac{1}{2} \int_{-\infty}^\infty e^{ikz}\big(e^{izt} + e^{-izt} \big)dz = \pi\big(\delta(t+k) + \delta(t-k) \big)$$ by the Fourier integral representation of the Dirac delta function. Only the latter delta function contributes since $t>0.$