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Asymptotic behaviour of the inflection points of Bessel functions. (English) Zbl 0719.33001
The inflection points in the title are ${j}_{\nu k}^{\text{'}\text{'}}$, the positive zeros of the second derivative of the Bessel function ${J}_{\nu }\left(z\right)$. Asymptotic expansions are derived for ${j}_{\nu k}^{\text{'}\text{'}}$ as $k\to \infty$ for fixed $\nu$ and as $\nu \to \infty$ for fixed k. Also derived is an asymptotic expansion of ${J}_{\nu }\left({j}_{\nu k}^{\text{'}\text{'}}\right)$ as $\nu \to \infty$. Finally, a lower bound of ${j}_{\nu k}^{\text{'}\text{'}}$ is derived for the larger k- values and values of $\nu$ satisfying $\nu \ge 7$, which implies that $|{J}_{\nu }\left({j}_{\nu k}^{\text{'}\text{'}}\right)|$ 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.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
zeros of Bessel function