# §10.24 Functions of Imaginary Order

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

 10.24.1 $x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d% }x}+(x^{2}+\nu^{2})w=0.$ ⓘ 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: §10.24, §10.24 Permalink: http://dlmf.nist.gov/10.24.E1 Encodings: TeX, pMML, png See also: Annotations for §10.24 and Ch.10

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

 10.24.2 $\displaystyle\widetilde{J}_{\nu}\left(x\right)$ $\displaystyle=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(J_{i% \nu}\left(x\right)\right),$ $\displaystyle\widetilde{Y}_{\nu}\left(x\right)$ $\displaystyle=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(Y_{i% \nu}\left(x\right)\right),$ ⓘ Defines: $\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)$: Bessel function of imaginary order and $\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)$: Bessel function of imaginary order Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{sech}\NVar{z}$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $\Re$: real part, $x$: real variable and $\nu$: complex parameter Referenced by: §10.24 Permalink: http://dlmf.nist.gov/10.24.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.24 and Ch.10

and

 10.24.3 $\Gamma\left(1+i\nu\right)=\left(\frac{\pi\nu}{\sinh\left(\pi\nu\right)}\right)% ^{\frac{1}{2}}e^{i\gamma_{\nu}},$

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

 10.24.4 $\displaystyle\widetilde{J}_{-\nu}\left(x\right)$ $\displaystyle=\widetilde{J}_{\nu}\left(x\right),$ $\displaystyle\widetilde{Y}_{-\nu}\left(x\right)$ $\displaystyle=\widetilde{Y}_{\nu}\left(x\right),$ ⓘ

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

 10.24.5 $\mathscr{W}\left\{\widetilde{J}_{\nu}\left(x\right),\widetilde{Y}_{\nu}\left(x% \right)\right\}=2/(\pi x).$

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

 10.24.6 $\displaystyle\widetilde{J}_{\nu}\left(x\right)$ $\displaystyle=\sqrt{2/(\pi x)}\cos\left(x-\tfrac{1}{4}\pi\right)+O\left(x^{-% \frac{3}{2}}\right),$ $\displaystyle\widetilde{Y}_{\nu}\left(x\right)$ $\displaystyle=\sqrt{2/(\pi x)}\sin\left(x-\tfrac{1}{4}\pi\right)+O\left(x^{-% \frac{3}{2}}\right).$

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

 10.24.7 $\displaystyle\widetilde{J}_{\nu}\left(x\right)$ $\displaystyle=\left(\frac{2\tanh\left(\frac{1}{2}\pi\nu\right)}{\pi\nu}\right)% ^{\frac{1}{2}}\cos\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{\nu}\right)+O% \left(x^{2}\right),$ 10.24.8 $\displaystyle\widetilde{Y}_{\nu}\left(x\right)$ $\displaystyle=\left(\frac{2\coth\left(\frac{1}{2}\pi\nu\right)}{\pi\nu}\right)% ^{\frac{1}{2}}\*\sin\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{\nu}\right)% +O\left(x^{2}\right),$ $\nu>0$,

and

 10.24.9 $\widetilde{Y}_{0}\left(x\right)=Y_{0}\left(x\right)=\frac{2}{\pi}\left(\ln% \left(\tfrac{1}{2}x\right)+\gamma\right)+O\left(x^{2}\ln x\right),$

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

In consequence of (10.24.6), when $x$ is large $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ 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 $\widetilde{J}_{\nu}\left(x\right)$ and $\tanh\left(\tfrac{1}{2}\pi\nu\right)\widetilde{Y}_{\nu}\left(x\right)$ or $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu\neq 0$ or $\nu=0$.

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

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