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On the computation of complex zeros of the modified Bessel function of the second kind. (English) Zbl 0594.34014
The author develops a high accuracy algorithm for computing the complex zeros of the modified Bessel function of the second kind ${K}_{\nu }\left(z\right)$ and its derivatives with respect to z in the case of real or complex values of $\nu$. He states that the implementation of the algorithm has proved its high efficiency. It is shown that all he first zeros of the function ${K}_{\nu +2p}\left(z\right)$, $p=0,1,···$, as $p\to \infty$, lie on a straight line parallel to the real axis, and the distance between two adjacent zeros tends to a constant. All the second, third,... zeros have the same asymptotic property. The author constructs a table of all the complex zeros with nine significant figures of the function ${K}_{\nu }\left(z\right)$ for $\nu =2\left(1\right)20$, and of the function ${K}_{\nu }^{\text{'}}\left(z\right)$ for $1/2\le \nu \le 20$. To determine the distribution of zeros of the derivatives of higher order, the loci of the zeros were constructed.
Reviewer: M.Shahin
##### MSC:
 34A40 Differential inequalities (ODE) 65D20 Computation of special functions, construction of tables
##### Keywords:
high accuracy algorithm