# §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 and Ch.10

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

 10.45.2 $\displaystyle\widetilde{I}_{\nu}\left(x\right)$ $\displaystyle=\Re\left(I_{i\nu}\left(x\right)\right),$ $\displaystyle\widetilde{K}_{\nu}\left(x\right)$ $\displaystyle=K_{i\nu}\left(x\right).$ ⓘ Defines: $\widetilde{I}_{\NVar{\nu}}\left(\NVar{x}\right)$: modified Bessel function of the first kind of imaginary order and $\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)$: modified Bessel function fo the second kind of imaginary order Symbols: $\mathrm{i}$: imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: 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 and Ch.10 Then 10.45.3 $\displaystyle\widetilde{I}_{-\nu}\left(x\right)$ $\displaystyle=\widetilde{I}_{\nu}\left(x\right),$ $\displaystyle\widetilde{K}_{-\nu}\left(x\right)$ $\displaystyle=\widetilde{K}_{\nu}\left(x\right),$

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

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

As $x\to+\infty$

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

As $x\to 0+$

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

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

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

when $\nu>0$, and

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

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

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

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

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