# §10.17 Asymptotic Expansions for Large Argument

## §10.17(i) Hankel’s Expansions

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

 10.17.1 $a_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k-1)^{2})}{% k!8^{k}}=\frac{{\left(\frac{1}{2}-\nu\right)_{k}}{\left(\frac{1}{2}+\nu\right)% _{k}}}{(-2)^{k}k!},$ $k\geq 1$, ⓘ Defines: $a_{k}(\nu)$: polynomial coefficient (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $k$: nonnegative integer and $\nu$: complex parameter Referenced by: §10.49(i), Erratum (V1.1.3) for Additions Permalink: http://dlmf.nist.gov/10.17.E1 Encodings: TeX, pMML, png Addition (effective with 1.1.3): An alternative Pochhammer symbol representation was added. See also: Annotations for §10.17(i), §10.17 and Ch.10
 10.17.2 $\omega=z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi,$ ⓘ Defines: $\omega$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.17.E2 Encodings: TeX, pMML, png See also: Annotations for §10.17(i), §10.17 and Ch.10

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

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

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

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

Corresponding expansions for other ranges of $\operatorname{ph}z$ 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 $b_{k}(\nu)=\frac{\left((4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k-3)^% {2})\right)(4\nu^{2}+4k^{2}-1)}{k!8^{k}}.$ ⓘ Defines: $b_{k}(\nu)$: polynomial coefficient (locally) Symbols: $!$: factorial (as in $n!$), $k$: nonnegative integer and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.17.E8 Encodings: TeX, pMML, png See also: Annotations for §10.17(ii), §10.17 and Ch.10

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

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

## §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 $\rselection{{H^{(1)}_{\nu}}\left(z\right)\\ {H^{(2)}_{\nu}}\left(z\right)}=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{% \pm i\omega}\left(\sum_{k=0}^{\ell-1}(\pm i)^{k}\frac{a_{k}(\nu)}{z^{k}}+R_{% \ell}^{\pm}(\nu,z)\right),$ $\ell=1,2,\dotsc$.

Then

 10.17.14 $\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm i% \infty}\left(t^{-\ell}\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,% \pm i\infty}\left(t^{-1}\right)\right),$ ⓘ Defines: $R_{\ell}^{\pm}(\nu,z)$: remainder (locally) Symbols: $\exp\NVar{z}$: exponential function, $\mathrm{i}$: imaginary unit, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_{k}(\nu)$: polynomial coefficient Referenced by: §10.17(iv), Erratum (V1.0.10) for Equation (10.17.14) Permalink: http://dlmf.nist.gov/10.17.E14 Encodings: TeX, pMML, png Errata (effective with 1.0.10): Originally the factor $\mathcal{V}_{z,\pm i\infty}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as $\mathcal{V}_{z,\pm i\infty}\left(t^{-\ell}\right)$. Reported 2014-09-27 by Gergő Nemes See also: Annotations for §10.17(iv), §10.17 and Ch.10

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

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

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

where $\Gamma\left(1-p,z\right)$ 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 $R_{\ell}^{\pm}(\nu,z)=(-1)^{\ell}2\cos\left(\nu\pi\right)\*\left(\sum_{k=0}^{m% -1}(\pm i)^{k}\frac{a_{k}(\nu)}{z^{k}}G_{\ell-k}\left(\mp 2iz\right)+R_{m,\ell% }^{\pm}(\nu,z)\right),$

where

 10.17.18 $R_{m,\ell}^{\pm}(\nu,z)=O\left(e^{-2|z|}z^{-m}\right),$ $|\operatorname{ph}\left(ze^{\mp\frac{1}{2}\pi i}\right)|\leq\pi$. ⓘ Defines: $R_{\ell}^{\pm}(\nu,z)$: remainder (locally) Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\operatorname{ph}$: phase, $m$: integer, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/10.17.E18 Encodings: TeX, pMML, png See also: Annotations for §10.17(v), §10.17 and Ch.10

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