Integral of the square of the Bessel Function

Question

Has anyone ever solved the integral of square of Bessel and Neumann functions? This is for standing wave analysis on a cylindrical object.

I need to integrate $(AJ(kr) + BY(kr))^2$ from $r=a$ to $r=b$, where $$A = \frac{-Y'(ka) V}{J'(ka)Y'(kb) - J'(kb)Y'(ka)}$$ and $$B= \frac{J'(ka) V}{J'(ka)Y'(kb) - J'(kb)Y'(ka)},$$ where $V$ is the velocity constant, and $a$ and $b$ are the radii of the torus.

$J(kr)$ is the Bessel function of the 1st kind and $Y(kr)$ is of the 2nd kind. $J'(\,\cdot\,)$ and $Y'(\,\cdot\,)$ are the first derivatives of these functions.

Thanks.

Answer 1

Are you sure you're not missing anything? Typically, in most analyses involving Bessels in cylindrical coordinates, integrals of their squares involve a Jacobian such as a factor of r as follows:

$$\int_{a}^{b} dx x J_1(k x)^2 $$

This is way easier to integrate. You get such a factor from conversion from rectangular to cylindrical coordinates and is required for conservation of energy considerations. Your integral, on the other hand, involves a nasty hypergeometric which gives little insight into the behavior of the result.