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10 Bessel FunctionsModified Bessel Functions

§10.42 Zeros

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Notes:
See Watson (1944, pp. 511–513) and Olver (1997b, p. 254).
Keywords:
modified Bessel functions
Referenced by:
Ch.10, Other Changes
Permalink:
http://dlmf.nist.gov/10.42
Addition (effective with 1.0.5):
The reference to Ferreira and Sesma (2008) has been added at the end of this section.

Reported 2012-07-30

Properties of the zeros of \mathop{I_{{\nu}}\/}\nolimits\!\left(z\right) and \mathop{K_{{\nu}}\/}\nolimits\!\left(z\right) may be deduced from those of \mathop{J_{{\nu}}\/}\nolimits\!\left(z\right) and \mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right), respectively, by application of the transformations (10.27.6) and (10.27.8).

For example, if \nu is real, then the zeros of \mathop{I_{{\nu}}\/}\nolimits\!\left(z\right) are all complex unless -2\ell<\nu<-(2\ell-1) for some positive integer \ell, in which event \mathop{I_{{\nu}}\/}\nolimits\!\left(z\right) has two real zeros.

The distribution of the zeros of \mathop{K_{{n}}\/}\nolimits\!\left(nz\right) in the sector -\tfrac{3}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{2}\pi in the cases n=1,5,10 is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle -\tfrac{1}{2}\pi so that in each case the cut lies along the positive imaginary axis. The zeros in the sector -\tfrac{1}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2}\pi are their conjugates.

\mathop{K_{{n}}\/}\nolimits\!\left(z\right) has no zeros in the sector |\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi; this result remains true when n is replaced by any real number \nu. For the number of zeros of \mathop{K_{{\nu}}\/}\nolimits\!\left(z\right) in the sector |\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi, when \nu is real, see Watson (1944, pp. 511–513).

For z-zeros of \mathop{K_{{\nu}}\/}\nolimits\!\left(z\right), with complex \nu, see Ferreira and Sesma (2008).

See also Kerimov and Skorokhodov (1984b, a).

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