10 Bessel FunctionsModified Bessel Functions10.41 Asymptotic Expansions for Large Order10.43 Integrals

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 ${H}_{\nu}^{\left(1\right)}\left(z\right)$, 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 $$ for some positive integer $\mathrm{\ell}$, 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 $-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{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 $-\frac{1}{2}\pi $ so that in each case the cut lies along the positive imaginary axis. The zeros in the sector $-\frac{1}{2}\pi \le \mathrm{ph}z\le \frac{3}{2}\pi $ are their conjugates.

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

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