# §10.45 Functions of Imaginary Order

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

 10.45.1 $x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d% }x}+(\nu^{2}-x^{2})w=0.$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $x$: real variable and $\nu$: complex parameter Referenced by: §10.45, §10.45 Permalink: http://dlmf.nist.gov/10.45.E1 Encodings: TeX, pMML, png See also: Annotations for 10.45

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

 10.45.2 $\displaystyle\mathop{\widetilde{I}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\Re{(\mathop{I_{i\nu}\/}\nolimits\!\left(x\right))},$ $\displaystyle\mathop{\widetilde{K}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{K_{i\nu}\/}\nolimits\!\left(x\right).$ Defines: $\mathop{\widetilde{I}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: modified Bessel function of the first kind of imaginary order and $\mathop{\widetilde{K}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: modified Bessel function fo the second kind of imaginary order Symbols: $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the first kind, $\mathop{K_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\Re{}$: real part, $x$: real variable and $\nu$: complex parameter Referenced by: §10.30(i), §10.74(viii) Permalink: http://dlmf.nist.gov/10.45.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.45 Then 10.45.3 $\displaystyle\mathop{\widetilde{I}_{-\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\widetilde{I}_{\nu}\/}\nolimits\!\left(x\right),$ $\displaystyle\mathop{\widetilde{K}_{-\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\widetilde{K}_{\nu}\/}\nolimits\!\left(x\right),$

and $\mathop{\widetilde{I}_{\nu}\/}\nolimits\!\left(x\right)$, $\mathop{\widetilde{K}_{\nu}\/}\nolimits\!\left(x\right)$ are real and linearly independent solutions of (10.45.1):

 10.45.4 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\widetilde{K}_{\nu}\/}\nolimits% \!\left(x\right),\mathop{\widetilde{I}_{\nu}\/}\nolimits\!\left(x\right)\right% \}=1/x.$

As $x\to+\infty$

 10.45.5 $\displaystyle\mathop{\widetilde{I}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=(2\pi x)^{-\frac{1}{2}}e^{x}\left(1+\mathop{O\/}\nolimits(x^{-1}% )\right),$ $\displaystyle\mathop{\widetilde{K}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=(\pi/(2x))^{\frac{1}{2}}e^{-x}\left(1+\mathop{O\/}\nolimits(x^{-% 1})\right).$

As $x\to 0+$

 10.45.6 $\mathop{\widetilde{I}_{\nu}\/}\nolimits\!\left(x\right)=\left(\frac{\mathop{% \sinh\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu}\right)^{\frac{1}{2}}\mathop{% \cos\/}\nolimits\!\left(\nu\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)% -\gamma_{\nu}\right)+\mathop{O\/}\nolimits\!\left(x^{2}\right),$

where $\gamma_{\nu}$ is as in §10.24. The corresponding result for $\mathop{\widetilde{K}_{\nu}\/}\nolimits\!\left(x\right)$ is given by

 10.45.7 $\mathop{\widetilde{K}_{\nu}\/}\nolimits\!\left(x\right)=-\left(\frac{\pi}{\nu% \mathop{\sinh\/}\nolimits\!\left(\pi\nu\right)}\right)^{\frac{1}{2}}\*\mathop{% \sin\/}\nolimits\!\left(\nu\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)% -\gamma_{\nu}\right)+\mathop{O\/}\nolimits\!\left(x^{2}\right),$

when $\nu>0$, and

 10.45.8 $\mathop{\widetilde{K}_{0}\/}\nolimits\!\left(x\right)=\mathop{K_{0}\/}% \nolimits\!\left(x\right)=-\mathop{\ln\/}\nolimits(\tfrac{1}{2}x)-\gamma+% \mathop{O\/}\nolimits(x^{2}\mathop{\ln\/}\nolimits x),$

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

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

For graphs of $\mathop{\widetilde{I}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\widetilde{K}_{\nu}\/}\nolimits\!\left(x\right)$ see §10.26(iii).

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