Title: Asymptotic relation for zeros of cross-product of Bessel functions and applications
Authors: Bobkov, Vladimír
Citation: BOBKOV, V. Asymptotic relation for zeros of cross-product of Bessel functions and applications. Journal of Mathematical Analysis and Applications, 2019, roč. 472, č. 1, s. 1078-1092. ISSN 0022-247X.
Issue Date: 2019
Publisher: Elsevier
Document type: článek
URI: 2-s2.0-85057886086
ISSN: 0022-247X
Keywords in different language: cross-product of Bessel functions;asymptotic of zeros;upper bound for zeros;Bessel functions;eigenvalues;Pleijel theorem
Abstract in different language: Let $a_{\nu,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_\nu(R z) Y_\nu(z) - J_\nu(z) Y_\nu(R z)$, where $\nu\geq 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $\alpha(x)$ characterizes the limit behavior of $a_{\nu,k}$ in the following sense: $$ \lim_{k \to \infty} \frac{a_{kx,k}}{k} = \alpha(x), \quad x \geq 0. $$ Moreover, we show that $$ a_{\nu,k} < \frac{\pi k}{R-1} + \frac{\pi \nu}{2 R}. $$ We use $\alpha(x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.
Rights: © Elsevier
Appears in Collections:Články / Articles (NTIS)

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