Asymptoptics for small argument of product of Bessel functions

Question

The problem: -The asymptotic expansion of the product of Bessel functions of different kind for small argument. I found the result that the following equation is finite for small argument, but I am not sure of this result. \begin{equation} \left[\alpha_{1}J_{m}^{'}(\alpha_{1}r)J_{m}(\alpha_{2}r)-\alpha_{2}J_{m}(\alpha_{1}r)J_{m}^{'}(\alpha_{2}r)\right]Y_{m}(\alpha_{2}r) \rightarrow 0 \end{equation} as $r\rightarrow 0$. The prime denotes derivative with respect to the argument.

I tried to solve it first by using the recurrence relations for the Bessel function of first kind, taking out the derivatives and then applying the asymptotic expansion for small argument, finding then that this equation goes to zero, so is finite at the origin. Any different approach and/or answer to this problem?

Answer 1

The expression to the left of the Bessel function of the second kind is zero:

\begin{equation} \left[\alpha_{1}J_{m}^{'}(\alpha_{1}r)J_{m}(\alpha_{2}r)-\alpha_{2}J_{m}(\alpha_{1}r)J_{m}^{'}(\alpha_{2}r)\right]=0 \end{equation} provided that $\alpha_{1}$ e $\alpha_{2}$ are different roots of:

• $J_{m}^{'}(\alpha_{1,2}r)=0$;
• $J_{m}(\alpha_{1,2}r)=0$;
• $ArJ_{m}^{'}(\alpha_{1,2}r)+BJ_{m}(\alpha_{1,2}r)=0$.

where A and B are independent of $r$.

See GRAY, Andrew et al. A treatise on Bessel functions and their applications to physics. Macmillan and Company, 1895, chapter VI, page 53.