# §10.42 Zeros

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).