An Integral invoving products of modified bessel functions

Question

I am a physicist working on a problem where the following integrals are concerned:

$$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$

$$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,$$

such that $\mu, p \in \mathbb{R}$, $l\in\mathbb{N}^+$, $I_\mu(k)$ is the modified bessel function of the first kind and $K_\mu(k)$ is that of the second kind.

When $l=0$ this integral has a closed form as can be found here. I wonder for integers $l\geq 1$, which should be a straightforward generalization, if it also admits a closed form.

Answer 1

A relevant integral formula can be found in paragraph 13.32 (Generalizations of Weber's second exponential integral) of

Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press (1922). ZBL50.0264.01.

for the bessel function $J_\alpha(x)$ of the first kind. Expressing the modified bessel functions $I_\alpha(x) = i^{-\alpha}J_\alpha(ix)$ and $K_\alpha(x) = \frac{\pi}{2}\frac{i^{-\alpha}J_\alpha(i x)-i^{\alpha}J_{-\alpha}(ix)}{\sin \alpha \pi}$, it provides the more general result

\begin{align} &\int_0^\infty k^{l+1}e^{-p^2k^2}I_\mu(k) K_\nu(k)\mathrm{d}k = \\ &\frac{\pi}{\sin \nu \pi} \frac{2^{-\mu +\nu -2} \Gamma \left(\frac{l+\mu -\nu +2}{2}\right) p^{-l-\mu +\nu -2}}{\Gamma (\mu +1) \Gamma (1-\nu )} \, _3F_3\left(\frac{\mu-\nu+1}{2},\frac{\mu-\nu+2}{2},\frac{l+\mu -\nu+2 }{2};\mu +1,1-\nu ,\mu -\nu +1;\frac{1}{p^2}\right)\\ &-\frac{\pi}{\sin \nu \pi}\frac{2^{-\mu -\nu -2} \Gamma \left(\frac{l+\mu +\nu +2}{2}\right) p^{-l-\mu -\nu -2} }{\Gamma (\mu +1) \Gamma (\nu +1)}\, _3F_3\left(\frac{\mu +\nu +1}{2},\frac{\mu +\nu +2}{2},\frac{l+\mu+\nu+2 }{2};\mu +1,\nu +1,\mu +\nu +1;\frac{1}{p^2}\right). \end{align} The second integral can be approached similarly.

Answer 2

for what it's worth, Mathematica gives a closed-form expression in terms of a hypergeometric function: _{$$\int_{0}^{\infty} k^{\ell+1} e^{-p^2k^2}(K_{\mu}(k))^2\,dk=$$ $$\mu^{-1}2^{-2 \mu-3} p^{-\ell-2 \mu-2} \left[\mu \Gamma (-\mu)^2 \Gamma \left(\frac{\ell}{2}+\mu+1\right) \, _2F_2\left(\mu+\frac{1}{2},\frac{\ell}{2}+\mu+1;\mu+1,2 \mu+1;\frac{1}{p^2}\right)+16^{\mu} \mu p^{4 \mu} \Gamma (\mu)^2 \Gamma \left(\frac{\ell}{2}-\mu+1\right) \, _2F_2\left(\frac{1}{2}-\mu,\frac{\ell}{2}-\mu+1;1-2 \mu,1-\mu;\frac{1}{p^2}\right)-4^\mu \pi l \Gamma \left(\frac{\ell}{2}\right) p^{2 \mu}\frac{1}{ \sin \pi \mu} \, _2F_2\left(\frac{1}{2},\frac{\ell}{2}+1;1-\mu,\mu+1;\frac{1}{p^2}\right)\right],$$} for $p>0$, $\ell>2|\mu|-2$.