I am trying to find an analytical expression (exact or approximate) for the following expression: $$ J_{n_1}(\alpha_1 x)= J_{n_2}(\alpha_2 x) $$ where $J_{n}(x)$ denotes a Bessel function, $n_1,n_2\in\mathbb{N}$ and $\alpha_1,\alpha_2\in\mathbb{R}$. This problem boils down to find the intersection of two Bessel functions which can be seen as either finding the zeros of their difference or an expression for their ratio. I haven't seen anything close that in the literature. Can you please help me solve this or point out to a paper that could help me solving it please? Thanks in advance!
2026-03-30 12:20:37.1774873237
Intersection of two Bessel functions
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