§11.2 Definitions

Principal values correspond to principal values of ${(\frac{1}{2}z)}^{\nu +1}$; compare §4.2(i).

The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of $z$. The functions $z^{-\nu -1}{𝐇}_{\nu }\left(z\right)$ and $z^{-\nu -1}{𝐋}_{\nu }\left(z\right)$ are entire functions of $z$ and $\nu$.

Principal values of ${𝐊}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$ correspond to principal values of the functions on the right-hand sides of (11.2.5) and (11.2.6).

Unless indicated otherwise, ${𝐇}_{\nu }\left(z\right)$, ${𝐊}_{\nu }\left(z\right)$, ${𝐋}_{\nu }\left(z\right)$, and ${𝐌}_{\nu }\left(z\right)$ assume their principal values throughout the DLMF.

In this subsection $A$ and $B$ are arbitrary constants.

When $z=x$, , and $\mathrm{\Re }\nu \ge 0$, numerically satisfactory general solutions of (11.2.7) are given by

(11.2.11) applies when $x$ is bounded, and (11.2.12) applies when $x$ is bounded away from the origin.

When $z\in ℂ$ and $\mathrm{\Re }\nu \ge 0$, numerically satisfactory general solutions of (11.2.7) are given by

(11.2.13) applies when $0\le \mathrm{ph}z\le \pi$ and $|z|$ is bounded. (11.2.14) applies when $-\pi \le \mathrm{ph}z\le 0$ and $|z|$ is bounded. (11.2.15) applies when $\left|\mathrm{ph}z\right|\le \pi$ and $z$ is bounded away from the origin.

When $\mathrm{\Re }\nu \ge 0$, numerically satisfactory general solutions of (11.2.9) are given by

(11.2.16) applies when $|\mathrm{ph}z|\le \frac{1}{2}\pi$ with $|z|$ bounded. (11.2.17) applies when $|\mathrm{ph}z|\le \frac{1}{2}\pi$ with $z$ bounded away from the origin.