- §10.25(i) Modified Bessel’s Equation
- §10.25(ii) Standard Solutions
- §10.25(iii) Numerically Satisfactory Pairs of Solutions

10.25.1 | $${z}^{2}\frac{{d}^{2}w}{{dz}^{2}}+z\frac{dw}{dz}-({z}^{2}+{\nu}^{2})w=0.$$ | ||

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.

10.25.2 | $${I}_{\nu}\left(z\right)={(\frac{1}{2}z)}^{\nu}\sum _{k=0}^{\mathrm{\infty}}\frac{{(\frac{1}{4}{z}^{2})}^{k}}{k!\mathrm{\Gamma}\left(\nu +k+1\right)}.$$ | ||

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 $\u2102\setminus (-\mathrm{\infty},0]$, 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

10.25.3 | $${K}_{\nu}\left(z\right)\sim \sqrt{\pi /(2z)}{\mathrm{e}}^{-z},$$ | ||

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 $\nu \in \u2102$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, 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 $.

Except where indicated otherwise it is assumed throughout the DLMF that the symbols ${I}_{\nu}\left(z\right)$ and ${K}_{\nu}\left(z\right)$ denote the principal values of these functions.

Corresponding to the symbol ${\mathcal{C}}_{\nu}$ introduced in §10.2(ii), we sometimes use ${\mathcal{Z}}_{\nu}\left(z\right)$ to denote ${I}_{\nu}\left(z\right)$, ${\mathrm{e}}^{\nu \pi \mathrm{i}}{K}_{\nu}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\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 ${I}_{-\nu}\left(z\right)$.

Pair | Region |
---|---|

${I}_{\nu}\left(z\right),{K}_{\nu}\left(z\right)$ | $|\mathrm{ph}z|\le \frac{1}{2}\pi $ |

${I}_{\nu}\left(z\right),{K}_{\nu}\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}}\right)$ | $\frac{1}{2}\pi \le \pm \mathrm{ph}z\le \frac{3}{2}\pi $ |