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On zeros of the modified Bessel function of the second kind. (English. Russian original) Zbl 1468.33009

Comput. Math. Math. Phys. 60, No. 5, 817-820 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 5, 837-840 (2020).
The zeros \(\nu_n\) of the modified Bessel function \(K_{i\nu}(x)\) at fixed argument \(x>0\) are shown to be countably infinite and simple. By a suitable transformation of the differential equation satisfied by \(K_{i\nu}(x)\) into a one-dimensional Schrodinger equation with an exponential potential, the equation is expressed as a boundary-value problem whose eigenvalues are the zeros \(\nu_n\).
By means of the standard quantisation rule the large-\(n\) behaviour of the zeros is shown to be given by \(\nu_n\sim\pi n/\log\,n\) as \(n\to\infty\).

MSC:

33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
Full Text: DOI

References:

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