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The answers to the question Bessel Functions Sturm-Liouville problem show that the eigenfunctions of the Helmholtz equation $$ (-\nabla^2+k^2)\psi(\mathbf r)=0 $$ in an annulus $a<r<b$, that have cylindrical separability of the form $\psi(r,\theta) = f(r)e^{im\theta}$ and which are subject to hard Dirichlet boundary conditions at both an inner and an outer ring, $$ f(a)=f(b)=0, $$ are linear combinations of Bessel functions of the first and second kinds, of the form $$ f(r) = N\bigg[Y_m(ka)J_m(kr)-J_m(ka)Y_m(kr)\bigg], $$ where the (square root of the) eigenvalue $k$ is fixed by requiring that the eigenfunction vanish at the outer ring: $$ f(b)/N= Y_m(ka)J_m(kb)-J_m(ka)Y_m(kb)=0 \tag{$*$} $$

I am looking for efficient and reliable ways to solve this equation, i.e. given $m$ and $b/a$, find the $k$ that corresponds to the eigenvalue.

For the circular problems, the zeroes of both $J_m(kr)$ and $Y_m(kr)$ are widely tabulated (example) and are often available as standard parts of scientific software packages (example), but the annular case isn't nearly as well-established. What are good ways to numerically solve that equation and reliably come up with the correct roots?

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    $\begingroup$ Ah. As often happens, once this post was mostly written, I found the relevant section in the DLMF; the asymptotics can then be used as initial seeds for a Newton's-method solver, providing sufficient reliability in finding the correct root for a large range of values of $b/a$, $m$ and the root order. I'll leave this up for anybody who wants to write it up and in case it's useful for someone else down the line. $\endgroup$ – E.P. Nov 23 '17 at 20:29
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    $\begingroup$ Hmmm. Upon a closer look, that's not quite enough. If someone has a reliable solver for getting the first few roots at highish $m$, say 5 or so, at $b/a \gtrsim 5€, that'd be mighty interesting. $\endgroup$ – E.P. Nov 24 '17 at 7:14

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