§10.24 Functions of Imaginary Order

With $z=x$ and $\nu$ replaced by $\mathrm{i}\nu$, Bessel’s equation (10.2.1) becomes

For $\nu \in ℝ$ and $x$ $\in (0,\mathrm{\infty })$ define

and

where ${\gamma }_{\nu }$ is real and continuous with ${\gamma }_0=0$; compare (5.4.3). Then

and ${\tilde{J}}_{\nu }\left(x\right)$, ${\tilde{Y}}_{\nu }\left(x\right)$ are linearly independent solutions of (10.24.1):

As $x\to +\mathrm{\infty }$, with $\nu$ fixed,

As $x\to 0+$, with $\nu$ fixed,

and

where $\gamma$ denotes Euler’s constant §5.2(ii).

In consequence of (10.24.6), when $x$ is large ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when $x$ is small either ${\tilde{J}}_{\nu }\left(x\right)$ and $\tanh \left(\frac{1}{2}\pi \nu \right){\tilde{Y}}_{\nu }\left(x\right)$ or ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu \ne 0$ or $\nu =0$.

For graphs of ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ see §10.3(iii).

For mathematical properties and applications of ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$, including zeros and uniform asymptotic expansions for large $\nu$, see Dunster (1990a). In this reference ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ are denoted respectively by $F_{\mathrm{i}\nu }(x)$ and $G_{\mathrm{i}\nu }(x)$.