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On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. (English) Zbl 0588.33005

The author obtains an asymptotic expansion for the lowest exceptional root of the equation

${j}^{\text{'}}{\phantom{\rule{-0.166667em}{0ex}}}_{\nu }\left(x\right){y}^{\text{'}}{\phantom{\rule{-0.166667em}{0ex}}}_{\nu }\left(\rho x\right)-{j}^{\text{'}}{\phantom{\rule{-0.166667em}{0ex}}}_{\nu }\left(\rho x\right){y}^{\text{'}}{\phantom{\rule{-0.166667em}{0ex}}}_{\nu }\left(x\right)$

where ${}^{\text{'}}=d/dx$ and ${j}_{\nu }$ and ${y}_{\nu }$ denote the spherical Bessel functions of the first and secind kind, respectively. The result is valid for $\rho$ $\to 1$, and it can be thought of as complementing that given by McMahon which is useful for larger zeros.

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