# §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 $x^{2}\frac{{d}^{2}w}{{dx}^{2}}+x\frac{dw}{dx}+(x^{2}+\nu^{2})w=0.$

For $\nu\in\Real$ and $x$ $\in(0,\infty)$ define

 10.24.2 $\displaystyle\mathop{\widetilde{J}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathrm{sech}\/}\nolimits\left(\tfrac{1}{2}\pi\nu\right)% \realpart{(\mathop{J_{i\nu}\/}\nolimits\!\left(x\right))},$ $\displaystyle\mathop{\widetilde{Y}_{\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathrm{sech}\/}\nolimits\left(\tfrac{1}{2}\pi\nu\right)% \realpart{(\mathop{Y_{i\nu}\/}\nolimits\!\left(x\right))},$

and

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

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

 10.24.4 $\displaystyle\mathop{\widetilde{J}_{-\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\widetilde{J}_{\nu}\/}\nolimits\!\left(x\right),$ $\displaystyle\mathop{\widetilde{Y}_{-\nu}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\widetilde{Y}_{\nu}\/}\nolimits\!\left(x\right),$

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

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

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

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

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

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

and

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

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

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

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

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