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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

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

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, \({}_0F_1\)
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