# How to evaluate the integral involving two Bessel functions and following elementary function?

Let's have the integral $$\int \limits_{0}^{\infty} J_{0}(kr) J_{0}(kr')\frac{kdk}{k^2 + a^2}.$$ How to evaluate it? I failed when was trying using integral representations for the Bessel functions.

• You want the answer or the technique? – Igor Rivin Dec 5 '13 at 17:01
• @IgorRivin : of course, the second, according to the title. – John Taylor Dec 5 '13 at 17:06

Let me first state the answer: for $r<r'$ $$\mathcal{I}(r,r',a)=\int_0^{\infty}J_0(kr)J_0(kr')\frac{kdk}{k^2+a^2}=I_0(ar)K_0(ar').\tag{1}$$ More general formulas can be found in Prudnikov-Brychkov-Marychev, Vol. 2.
Thus we see that $\mathcal{I}(r,r',a)$ solves the modified Bessel equation w.r.t. $ar$ and should therefore be a linear combination of its two independent solutions. Moreover since it is regular at $r=0$, it can only be proportional to $I_0(ar)$.
Similarly, $\mathcal{I}(r,r',a)$ solves the modified Bessel equation w.r.t. $ar'$ and vanishes at infinity, hence it can only be proportional to $K_0(ar')$. Therefore, $$\mathcal{I}(r,r',a)=\operatorname{const}\cdot I_0(ar) K_0(ar').$$ The remaining constant can be fixed in several ways - e.g. using the known asymptotic behavior of Bessel functions.