§10.45 Functions of Imaginary Order

With $z=x$, and $\nu$ replaced by $\mathrm{i}\nu$, the modified Bessel’s equation (10.25.1) becomes

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

and ${\tilde{I}}_{\nu }\left(x\right)$, ${\tilde{K}}_{\nu }\left(x\right)$ are real and linearly independent solutions of (10.45.1):

As $x\to +\mathrm{\infty }$

As $x\to 0+$

where ${\gamma }_{\nu }$ is as in §10.24. The corresponding result for ${\tilde{K}}_{\nu }\left(x\right)$ is given by

when $\nu >0$, and

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

In consequence of (10.45.5)–(10.45.7), ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either ${\tilde{I}}_{\nu }\left(x\right)$ and $(1/\pi )\sinh \left(\pi \nu \right){\tilde{K}}_{\nu }\left(x\right)$, or ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu \ne 0$ or $\nu =0$.

For graphs of ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ see §10.26(iii).

For properties of ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). In this reference ${\tilde{I}}_{\nu }\left(x\right)$ is denoted by $(1/\pi )\sinh \left(\pi \nu \right)L_{\mathrm{i}\nu }(x)$. See also Gil et al. (2003a), Balogh (1967) and Booker et al. (2013).