Kerimov, M. K.; Skorokhodov, S. L. On the computation of the complex zeros of the Bessel functions \(J_{\nu}(z)\) and \(I_{\nu}(z)\) and their derivatives. (Russian) Zbl 0561.65011 Zh. Vychisl. Mat. Mat. Fiz. 24, No. 10, 1497-1513 (1984). The authors compute complex zeros of \(J_{\nu}(z)\), \(I_{\nu}(z)\) and their derivatives, where \(\nu\) is real. It is known that if \(\nu >-1\), then \(J_{\nu}(z)\) has only real zeros, while if \(\nu <-1\), and \(\nu\) is not an integer, then \(J_{\nu}(z)\) has \(2\lfloor -\nu \rfloor\) complex zeros. The method is due to Newton iteration, starting from an asymptotic formula giving the approximate zeros. They give the table of complex zeros of \(I_{\nu}(z)\) for \(\nu =1.5(1)20.5\) and of \(I'_{\nu}(z)\) for \(\nu =0.5(1)20.5\). Further they consider the double zero of \(J_{\nu}(z)\). According to the Bessel’s differential equation, \(J'_{\nu}(z_ 0) = J''_{\nu}(z_ 0) = 0\) may occur only at \(z_0 = \mp\nu\), \(\nu<0\). They give first 100 values of such \(\nu\)’s. The difference of contiguous two values tends to 1. Reviewer: S.Hitotumatu Cited in 3 ReviewsCited in 1 Document MSC: 65D20 Computation of special functions and constants, construction of tables 65H05 Numerical computation of solutions to single equations 33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\) Keywords:complex zero of Bessel functions; derivative of Bessel functions; Newton iteration; asymptotic formula PDFBibTeX XMLCite \textit{M. K. Kerimov} and \textit{S. L. Skorokhodov}, Zh. Vychisl. Mat. Mat. Fiz. 24, No. 10, 1497--1513 (1984; Zbl 0561.65011) Digital Library of Mathematical Functions: §10.21(ix) Complex Zeros ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions Complex Zeros ‣ §10.74(vi) Zeros and Associated Values ‣ §10.74 Methods of Computation ‣ Computation ‣ Chapter 10 Bessel Functions 4th item ‣ §10.75(vi) Zeros of Modified Bessel Functions and their Derivatives ‣ §10.75 Tables ‣ Computation ‣ Chapter 10 Bessel Functions