I wish to calculate the following sum,
$$A=\sum_{\nu=-\infty}^{\infty}J_{\nu}(\xi)\delta(\nu-a)$$
where, $\xi=\beta\sin(\theta)\sin(\phi)$ and the sum needs to be calculated when $\beta\gg 1$.
I find in the literature that due to the limit on $\beta$ the sum could be converted to an integral over $\nu$ and the $\delta$ function would give,
$$A\approx J_{a}(\xi)$$
After this, the literature further evaluates the asymptotic value for $J_{a}(\xi)$ following two limits. While $a^2>\xi^2$, the asymptotic value of $J_{a}(\xi)$ is said to be evaluated from $K_{1/3}$, with $K_{1/3}$ being the modified Bessel function, on the other hand, if $a^2<\xi^2$ the asymptotic value said to be calculated from a combination of $J_{1/3}$ and $J_{-1/3}$.
I could not find a proper derivation of the calculation discussed above. It would be great to have some help in this regard.
Best Answer
I belive it refers to the approximation of $J_a(\xi)$ in the transition region. Assume that $a \gg 0$. Nicholson found that $$ J_a (\xi ) \approx \frac{1}{\pi }\sqrt {\frac{{2(a - \xi )}}{{3a}}} K_{1/3}\! \left( {\frac{{2^{3/2} (a - \xi )^{3/2} }}{{3a^{1/2} }}} \right) = \frac{{2^{1/3} }}{{a^{1/3} }}\operatorname{Ai}\left( {\frac{{2^{1/3} (a - \xi )}}{{a^{1/3} }}} \right) $$ for $a\geq \xi$, and \begin{align*} J_a (\xi ) & \approx \frac{1}{3}\sqrt {\frac{{2(\xi - a)}}{a}} \left[ {J_{ 1/3} \!\left( {\frac{{2^{3/2} (\xi - a)^{3/2} }}{{3a^{1/2} }}} \right) + J_{ - 1/3}\! \left( {\frac{{2^{3/2} (\xi - a)^{3/2} }}{{3a^{1/2} }}} \right)} \right] \\ & = \frac{{2^{1/3} }}{{a^{1/3} }}\operatorname{Ai}\left( {\frac{{2^{1/3} (a - \xi )}}{{a^{1/3} }}} \right) \end{align*} for $a\leq \xi$. Here $\operatorname{Ai}$ denotes the Airy function. These formulae are discussed in G. N. Watson's book A Treatise on the Theory of Bessel Functions ($\S\S$8.43 starting on page 248). Watson does not discuss it, but these approximations are useful only when $\left| {\xi - a} \right| = o(\left| a \right|^{9/15} )$. This condition follows from the more general result due to F. W. J. Olver. If you want approximations that are valid in larger domains, you can use the uniform asymptotic expansion of $J_a(\xi)$.