10 Bessel Functions Modified Bessel Functions 10.41 Asymptotic Expansions for Large Order 10.43 Integrals

§10.42 Zeros

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Keywords:: modified Bessel functions, zeros
Notes:: See Watson (1944, pp. 511–513) and Olver (1997b, p. 254).
Referenced by:: Ch.10, Erratum (V1.0.5) for References
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.
See also:: Annotations for Ch.10

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

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

The distribution of the zeros of $K_{n}\left(nz\right)$ in the sector $-\tfrac{3}{2}\pi\leq\operatorname{ph}z\leq\tfrac{1}{2}\pi$ in the cases 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 $-\tfrac{1}{2}\pi\leq\operatorname{ph}z\leq\tfrac{3}{2}\pi$ are their conjugates.

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

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

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

10.41 Asymptotic Expansions for Large Order 10.43 Integrals

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