# §10.17(i) Hankel’s Expansions

Define $a_{0}(\nu)=1$,

 10.17.1 $a_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k-1)^{2})}{% k!8^{k}},$ $k\geq 1$, Symbols: $!$: $n!$: factorial, $k$: nonnegative integer, $\nu$: complex parameter and $a_{k}(\nu)$: expansion Referenced by: §10.49(i) Permalink: http://dlmf.nist.gov/10.17.E1 Encodings: TeX, pMML, png
 10.17.2 $\omega=z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi,$ Symbols: $z$: complex variable, $\nu$: complex parameter and $\omega$ Permalink: http://dlmf.nist.gov/10.17.E2 Encodings: TeX, pMML, png

and let $\delta$ denote an arbitrary small positive constant. Then as $z\to\infty$, with $\nu$ fixed,

 10.17.3 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{% \frac{1}{2}}\*\left(\mathop{\cos\/}\nolimits\omega\sum_{k=0}^{\infty}(-1)^{k}% \frac{a_{2k}(\nu)}{z^{2k}}-\mathop{\sin\/}\nolimits\omega\sum_{k=0}^{\infty}(-% 1)^{k}\frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$,
 10.17.4 $\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{% \frac{1}{2}}\*\left(\mathop{\sin\/}\nolimits\omega\sum_{k=0}^{\infty}(-1)^{k}% \frac{a_{2k}(\nu)}{z^{2k}}+\mathop{\cos\/}\nolimits\omega\sum_{k=0}^{\infty}(-% 1)^{k}\frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$,
 10.17.5 $\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)\sim\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}e^{i\omega}\sum_{k=0}^{\infty}i^{k}\frac{a_{k}(\nu)}{z^{k% }},$ $-\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq 2\pi-\delta$,
 10.17.6 $\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)\sim\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}e^{-i\omega}\sum_{k=0}^{\infty}(-i)^{k}\frac{a_{k}(\nu)}{% z^{k}},$ $-2\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi-\delta$,

where the branch of $z^{\frac{1}{2}}$ is determined by

 10.17.7 $z^{\frac{1}{2}}=\mathop{\exp\/}\nolimits\left(\tfrac{1}{2}\mathop{\ln\/}% \nolimits|z|+\tfrac{1}{2}i\mathop{\mathrm{ph}\/}\nolimits z\right).$

Corresponding expansions for other ranges of $\mathop{\mathrm{ph}\/}\nolimits z$ can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4).

# §10.17(ii) Asymptotic Expansions of Derivatives

We continue to use the notation of §10.17(i). Also, $b_{0}(\nu)=1$, $b_{1}(\nu)=(4\nu^{2}+3)/8$, and for $k\geq 2$,

 10.17.8 $b_{k}(\nu)=\frac{\left((4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k-3)^% {2})\right)(4\nu^{2}+4k^{2}-1)}{k!8^{k}}.$ Symbols: $!$: $n!$: factorial, $k$: nonnegative integer, $\nu$: complex parameter and $b_{k}(\nu)$: expansion Permalink: http://dlmf.nist.gov/10.17.E8 Encodings: TeX, pMML, png

Then as $z\to\infty$ with $\nu$ fixed,

 10.17.9 $\displaystyle{\mathop{J_{\nu}\/}\nolimits^{\prime}}\!\left(z\right)$ $\displaystyle\sim-\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\left(\mathop{\sin% \/}\nolimits\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{b_{2k}(\nu)}{z^{2k}}+% \mathop{\cos\/}\nolimits\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{b_{2k+1}(\nu)}{% z^{2k+1}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$, 10.17.10 $\displaystyle{\mathop{Y_{\nu}\/}\nolimits^{\prime}}\!\left(z\right)$ $\displaystyle\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\left(\mathop{\cos% \/}\nolimits\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{b_{2k}(\nu)}{z^{2k}}-% \mathop{\sin\/}\nolimits\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{b_{2k+1}(\nu)}{% z^{2k+1}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$,
 10.17.11 $\displaystyle{\mathop{{H^{(1)}_{\nu}}\/}\nolimits^{\prime}}\!\left(z\right)$ $\displaystyle\sim i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{i\omega}\sum_{% k=0}^{\infty}i^{k}\frac{b_{k}(\nu)}{z^{k}},$ $-\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq 2\pi-\delta$, 10.17.12 $\displaystyle{\mathop{{H^{(2)}_{\nu}}\/}\nolimits^{\prime}}\!\left(z\right)$ $\displaystyle\sim-i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{-i\omega}\sum_% {k=0}^{\infty}(-i)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $-2\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi-\delta$.

# §10.17(iii) Error Bounds for Real Argument and Order

In the expansions (10.17.3) and (10.17.4) assume that $\nu\geq 0$ and $z>0$. Then the remainder associated with the sum $\sum_{k=0}^{\ell-1}(-1)^{k}a_{2k}(\nu)z^{-2k}$ does not exceed the first neglected term in absolute value and has the same sign provided that $\ell\geq\max(\tfrac{1}{2}\nu-\tfrac{1}{4},1)$. Similarly for $\sum_{k=0}^{\ell-1}(-1)^{k}a_{2k+1}(\nu)z^{-2k-1}$, provided that $\ell\geq\max(\tfrac{1}{2}\nu-\tfrac{3}{4},1)$.

In the expansions (10.17.5) and (10.17.6) assume that $\nu>-\tfrac{1}{2}$ and $z>0$. If these expansions are terminated when $k=\ell-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\ell\geq\max(\nu-\tfrac{1}{2},1)$.

# §10.17(iv) Error Bounds for Complex Argument and Order

For (10.17.5) and (10.17.6) write

 10.17.13 $\rselection{\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)\\ \mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)}=\left(\frac{2}{\pi z}% \right)^{\frac{1}{2}}e^{\pm i\omega}\left(\sum_{k=0}^{\ell-1}(\pm i)^{k}\frac{% a_{k}(\nu)}{z^{k}}+R_{\ell}^{\pm}(\nu,z)\right),$ $\ell=1,2,\ldots$.

Then

 10.17.14 $\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathop{\mathcal{V}_{z,% \pm i\infty}\/}\nolimits\!\left(t^{-\ell}\right)\*\mathop{\exp\/}\nolimits% \left(|\nu^{2}-\tfrac{1}{4}|\mathop{\mathcal{V}_{z,\pm i\infty}\/}\nolimits\!% \left(t^{-\ell}\right)\right),$ Symbols: $\mathop{\exp\/}\nolimits z$: exponential function, $\mathop{\mathcal{V}_{a,b}\/}\nolimits\!\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter, $a_{k}(\nu)$: expansion and $R_{\ell}^{\pm}(\nu,z)$: remainder Referenced by: §10.17(iv) Permalink: http://dlmf.nist.gov/10.17.E14 Encodings: TeX, pMML, png

where $\mathop{\mathcal{V}\/}\nolimits$ denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that $|\imagpart{t}|$ changes monotonically. Bounds for $\mathop{\mathcal{V}_{z,i\infty}\/}\nolimits\!\left(t^{-\ell}\right)$ are given by

 10.17.15 $\mathop{\mathcal{V}_{z,i\infty}\/}\nolimits\!\left(t^{-\ell}\right)\leq\begin{% cases}|z|^{-\ell},&0\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi,\\ \chi(\ell)|z|^{-\ell},&\parbox[t]{224.037pt}{-\tfrac{1}{2}\pi\leq\mathop{% \mathrm{ph}\/}\nolimits z\leq 0 or \pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2}\pi,}\\ 2\chi(\ell)|\imagpart{z}|^{-\ell},&\parbox[t]{224.037pt}{-\pi<\mathop{\mathrm% {ph}\/}\nolimits z\leq-\tfrac{1}{2}\pi or \tfrac{3}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z<2\pi,}\end{cases}$

where $\chi(\ell)=\pi^{\frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\ell% +1\right)/\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\ell+\tfrac{1}{2}\right)$; see §9.7(i). The bounds (10.17.15) also apply to $\mathop{\mathcal{V}_{z,-i\infty}\/}\nolimits\!\left(t^{-\ell}\right)$ in the conjugate sectors.

Corresponding error bounds for (10.17.3) and (10.17.4) are obtainable by combining (10.17.13) and (10.17.14) with (10.4.4).

# §10.17(v) Exponentially-Improved Expansions

As in §9.7(v) denote

 10.17.16 $\mathop{G_{p}\/}\nolimits\!\left(z\right)=\frac{e^{z}}{2\pi}\mathop{\Gamma\/}% \nolimits\!\left(p\right)\mathop{\Gamma\/}\nolimits\!\left(1-p,z\right),$

where $\mathop{\Gamma\/}\nolimits\!\left(1-p,z\right)$ is the incomplete gamma function (§8.2(i)). Then in (10.17.13) as $z\to\infty$ with $|\ell-2|z||$ bounded and $m$ ($\geq 0$) fixed,

 10.17.17 $R_{\ell}^{\pm}(\nu,z)=(-1)^{\ell}2\mathop{\cos\/}\nolimits(\nu\pi)\*\left(\sum% _{k=0}^{m-1}(\pm i)^{k}\frac{a_{k}(\nu)}{z^{k}}\mathop{G_{\ell-k}\/}\nolimits% \!\left(\mp 2iz\right)+R_{m,\ell}^{\pm}(\nu,z)\right),$

where

 10.17.18 $R_{m,\ell}^{\pm}(\nu,z)=\mathop{O\/}\nolimits\!\left(e^{-2|z|}z^{-m}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits(ze^{\mp\frac{1}{2}\pi i})|\leq\pi$. Symbols: $\mathop{O\/}\nolimits\!\left(x\right)$: order not exceeding, $e$: base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$: phase, $m$: integer, $z$: complex variable, $\nu$: complex parameter and $R_{\ell}^{\pm}(\nu,z)$: remainder Permalink: http://dlmf.nist.gov/10.17.E18 Encodings: TeX, pMML, png

For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).