§10.25 Definitions

This equation is obtained from Bessel’s equation (10.2.1) on replacing $z$ by $\pm \mathrm{i}z$, and it has the same kinds of singularities. Its solutions are called modified Bessel functions or Bessel functions of imaginary argument.

This solution has properties analogous to those of $J_{\nu }\left(z\right)$, defined in §10.2(ii). In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$.

The defining property of the second standard solution $K_{\nu }\left(z\right)$ of (10.25.1) is

as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$ . It has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$.

Both $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ are real when $\nu$ is real and $\mathrm{ph}z=0$.

For fixed $z$ $(\ne 0)$ each branch of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ is entire in $\nu$.

Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). It is assumed that $\mathrm{\Re }\nu \ge 0$. When , $I_{\nu }\left(z\right)$ is replaced by $I_{-\nu }\left(z\right)$.