§10.2 Definitions

This solution of (10.2.1) is an analytic function of $z\in ℂ$, except for a branch point at $z=0$ when $\nu$ is not an integer. The principal branch of $J_{\nu }\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty },0]$.

When $\nu =n$ $(\in \mathbb{Z})$, $J_{\nu }\left(z\right)$ is entire in $z$.

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

When $\nu$ is an integer the right-hand side is replaced by its limiting value:

Whether or not $\nu$ is an integer $Y_{\nu }\left(z\right)$ has a branch point at $z=0$. The principal branch corresponds to the principal branches of $J_{\pm \nu }\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$.

Except in the case of $J_{\pm n}\left(z\right)$, the principal branches of $J_{\nu }\left(z\right)$ and $Y_{\nu }\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$; compare §4.2(i).

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

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

These solutions of (10.2.1) are denoted by $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$, and their defining properties are given by

as $z\to \mathrm{\infty }$ in $-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$, and

as $z\to \mathrm{\infty }$ in $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$, where $\delta$ is an arbitrary small positive constant. Each solution has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$.

The principal branches of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$.

For fixed $z$ $(\ne 0)$ each branch of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ is entire in $\nu$.

Except where indicated otherwise, it is assumed throughout the DLMF that the symbols $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ denote the principal values of these functions.

The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$.