Bessel and Neumann functions:ordering the zeroes

Question

I realize the abundance of the literature on this theme which is a disservice in this case, because there are too many formulas in all the books I've found and no explicit answers. Maybe someone could give me a good reference, the question is quiet classical.

Let us consider a Bessel equation $$ r^2 R'' + r R' + (r-k^2) R = 0 $$

with the conditions on the boundaries $R(\sqrt{\lambda})=R(2\sqrt{\lambda})=0$. This comes from the equation of oscillation of an annulus with fixed boundaries. The question is to find the frequencies of oscillation in an increasing order, that is just find such $\sqrt{\lambda}$ that the system above holds.

We know that the general solution of the equation is the linear combination between Bessel and Neyman functions $J_k$ and $Y_k$, $k \in \mathbb{N}\cup \{0\}$ so the question is reduced to ordering the zeroes of $J_k$ and $Y_k$. Is there an answer in a form of explicit sequence or, at least, how many first terms do we know?

Answer 1

This is just to mark this question as answered, as the comments were enough.

Zeros of Bessel functions are the subject of Chapter XV in Watson's Treatise on the theory of Bessel Functions (1922).

Many older results are found there, together with new ones (at the time).

It seems that what you were asking is the following (page 487):

• $j_k$ is the $k$-th strictly positive real zero of the Bessel function of the first kind $J_k$,
• $j^\prime_k$ is the $k$-th positive real zero of the derivative of $J_k$
• $y_k$ is the $k$-th positive real zero of the Bessel function of the second kind $Y_k$, then

$...<j_{k-1} < j^\prime_{k} < y_{k} <j_{k}<...$

The reason of this exclusion of $0$, which is the first zero for all $J_n$, $n\geq1$, is that otherwise two different rules should be given for $n=0$ and otherwise. Watson attributes this result to Schafheitlin, Journal für Math. CXXII, 1900, pp. 317-321.