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Asymptotic behaviour of the inflection points of Bessel functions. (English) Zbl 0719.33001
The inflection points in the title are $j''\sb{\nu k}$, the positive zeros of the second derivative of the Bessel function $J\sb{\nu}(z)$. Asymptotic expansions are derived for $j''\sb{\nu k}$ as $k\to \infty$ for fixed $\nu$ and as $\nu\to \infty$ for fixed k. Also derived is an asymptotic expansion of $J\sb{\nu}(j''\sb{\nu k})$ as $\nu\to \infty$. Finally, a lower bound of $j''\sb{\nu k}$ is derived for the larger k- values and values of $\nu$ satisfying $\nu\ge 7$, which implies that $\vert J\sb{\nu}(j''\sb{\nu k})\vert$ is decreasing from a certain value of k. The results are based on asymptotic expansions of the Bessel functions with error bounds, as derived by Olver.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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