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On multiple zeros of derivatives of Bessel’s cylindrical functions. (English) Zbl 0611.33010
The authors discuss double zeros of the functions ${J}_{\nu }^{\left(s\right)}\left(z\right)$, ${Y}_{\nu }^{\left(s\right)}\left(z\right)$, $s=1,2,3$, where $\nu$ is real and z is complex. In the case $s=0$, there are no double zeros except possibly at the singular points 0, $\infty$ of the Bessel differential equation. For $s=1$, such ”nonsingular” zeros must lie on the curves $z=±\nu$. The authors find similar curves in the cases $s=2,3$ and hence find asymptotic formulas for these possible zeros as $\nu \to ±\infty$. They summarize the results of other investigations and computations. More details are to be found in the authors’ paper in Zh. Vychisl. Mat. Mat. Fiz. 25, No.12, 1749-1760 (1985; Zbl 0588.65015). A typical result is that there is a single ${\nu }_{n}$ on each interval (-n-1,-n), $n=1,2,···$, for which ${J}_{{\nu }_{n}}^{\text{'}}\left(±z\right)$ has a double zero and $-{\nu }_{n}=n+1/6+o\left(1\right),$ $n\to \infty$. The interest in such double zeros arose in work of J. Lense in the 1930’s on conformal mappings implemented by Bessel functions and in recent work by the authors on the investigation and computation of complex zeros of cylinder functions.
Reviewer: M.E.Muldoon
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 65D20 Computation of special functions, construction of tables 30C15 Zeros of polynomials, etc. (one complex variable)
##### Keywords:
multiple zeros; Bessel functions; cylinder functions