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Zbl 0561.65011
Kerimov, M.K.; Skorokhodov, S.L.
On the computation of the complex zeros of the Bessel functions $J\sb{\nu}(z)$ and $I\sb{\nu}(z)$ and their derivatives.
(Russian)
[J] Zh. Vychisl. Mat. Mat. Fiz. 24, No.10, 1497-1513 (1984). ISSN 0044-4669

The authors compute complex zeros of $J\sb{\nu}(z)$, $I\sb{\nu}(z)$ and their derivatives, where $\nu$ is real. It is known that if $\nu >-1$, then $J\sb{\nu}(z)$ has only real zeros, while if $\nu <-1$, and $\nu$ is not an integer, then $J\sb{\nu}(z)$ 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 $I\sb{\nu}(z)$ for $\nu =1.5(1)20.5$ and of $I'\sb{\nu}(z)$ for $\nu =0.5(1)20.5$. Further they consider the double zero of $J\sb{\nu}(z)$. According to the Bessel's differential equation, $J'\sb{\nu}(z\sb 0) = J''\sb{\nu}(z\sb 0) = 0$ may occur only at $z\sb0 = \mp\nu$, $\nu<0$. They give first 100 values of such $\nu$'s. The difference of contiguous two values tends to 1.
[S.Hitotumatu]
MSC 2000:
*65D20 Computation of special functions
65H05 Single nonlinear equations (numerical methods)
33C10 Cylinder functions, etc.

Keywords: complex zero of Bessel functions; derivative of Bessel functions; Newton iteration; asymptotic formula

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