§10.17 Asymptotic Expansions for Large Argument

Define $a_0(\nu )=1$,

10.17.1		$$a_k(\nu )=\frac{(4{\nu }^2-1^2)(4{\nu }^2-3^2)\mathrm{\cdots }(4{\nu }^2-{(2k-1)}^2)}{k!8^k},$$
$k\ge 1$,
ⓘ Defines: $a_k(\nu )$: polynomial coefficient (locally) Symbols: $!$: factorial (as in $n!$), $k$: nonnegative integer and $\nu$: complex parameter Referenced by: §10.49(i) Permalink: http://dlmf.nist.gov/10.17.E1 Encodings: TeX, pMML, png See also: Annotations for §10.17(i), §10.17 and Ch.10

10.17.2		$$\omega =z-\frac{1}{2}\nu \pi -\frac{1}{4}\pi ,$$
ⓘ 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 $z\to \mathrm{\infty }$, with $\nu$ fixed,

10.17.3		$$J_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}-\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$
$|\mathrm{ph}z|\le \pi -\delta$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$ A&S Ref: 9.2.5 Referenced by: §10.17(i), §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.49(i), §10.61(v), §10.7(ii) Permalink: http://dlmf.nist.gov/10.17.E3 Encodings: TeX, pMML, png See also: Annotations for §10.17(i), §10.17 and Ch.10

10.17.4		$$Y_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$
$|\mathrm{ph}z|\le \pi -\delta$,
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$ A&S Ref: 9.2.6 Referenced by: §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.7(ii), §11.6(i) Permalink: http://dlmf.nist.gov/10.17.E4 Encodings: TeX, pMML, png See also: Annotations for §10.17(i), §10.17 and Ch.10

10.17.5		$$H_{\nu }^{(1)}\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\omega }\sum _{k=0}^{\mathrm{\infty }}{\mathrm{i}}^k\frac{a_k(\nu )}{z^k},$$
$-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$,
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$ A&S Ref: 9.2.7 Referenced by: §10.17(i), §10.17(iii), §10.17(iv), §10.41(v) Permalink: http://dlmf.nist.gov/10.17.E5 Encodings: TeX, pMML, png See also: Annotations for §10.17(i), §10.17 and Ch.10

10.17.6		$$H_{\nu }^{(2)}\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}\omega }\sum _{k=0}^{\mathrm{\infty }}{(-\mathrm{i})}^k\frac{a_k(\nu )}{z^k},$$
$-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$,
ⓘ Symbols: $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$ A&S Ref: 9.2.8 Referenced by: §10.17(i), §10.17(iii), §10.17(iv), §10.41(v), §10.49(i), §10.61(v) Permalink: http://dlmf.nist.gov/10.17.E6 Encodings: TeX, pMML, png See also: Annotations for §10.17(i), §10.17 and Ch.10

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

10.17.7		$$z^{\frac{1}{2}}=\exp \left(\frac{1}{2}\ln |z|+\frac{1}{2}\mathrm{i}\mathrm{ph}z\right).$$
ⓘ Symbols: $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable Permalink: http://dlmf.nist.gov/10.17.E7 Encodings: TeX, pMML, png See also: Annotations for §10.17(i), §10.17 and Ch.10

Corresponding expansions for other ranges of $\mathrm{ph}z$ 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).

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 $k\ge 2$,

10.17.8		$$b_k(\nu )=\frac{\left((4{\nu }^2-1^2)(4{\nu }^2-3^2)\mathrm{\cdots }(4{\nu }^2-{(2k-3)}^2)\right)(4{\nu }^2+4k^2-1)}{k!8^k}.$$
ⓘ 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 $z\to \mathrm{\infty }$ with $\nu$ fixed,

10.17.9		$J_{\nu }^{\prime }\left(z\right)$	$\sim -{\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\left(\sin \omega {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{b_{2k}(\nu )}{z^{2k}}}+\cos \omega {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{b_{2k+1}(\nu )}{z^{2k+1}}}\right),$
$|\mathrm{ph}z|\le \pi -\delta$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient A&S Ref: 9.2.11 Permalink: http://dlmf.nist.gov/10.17.E9 Encodings: TeX, pMML, png See also: Annotations for §10.17(ii), §10.17 and Ch.10
10.17.10		$Y_{\nu }^{\prime }\left(z\right)$	$\sim {\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\left(\cos \omega {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{b_{2k}(\nu )}{z^{2k}}}-\sin \omega {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{b_{2k+1}(\nu )}{z^{2k+1}}}\right),$
$|\mathrm{ph}z|\le \pi -\delta$,
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient A&S Ref: 9.2.12 Permalink: http://dlmf.nist.gov/10.17.E10 Encodings: TeX, pMML, png See also: Annotations for §10.17(ii), §10.17 and Ch.10

10.17.11		$H_{\nu }^{(1)}{}^{\prime }\left(z\right)$	$\sim \mathrm{i}{\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\omega }{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\mathrm{i}}^k{\displaystyle \frac{b_k(\nu )}{z^k}},$
$-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$,
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient A&S Ref: 9.2.13 Referenced by: §10.18(iii) Permalink: http://dlmf.nist.gov/10.17.E11 Encodings: TeX, pMML, png See also: Annotations for §10.17(ii), §10.17 and Ch.10
10.17.12		$H_{\nu }^{(2)}{}^{\prime }\left(z\right)$	$\sim -\mathrm{i}{\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}\omega }{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-\mathrm{i})}^k{\displaystyle \frac{b_k(\nu )}{z^k}},$
$-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$.
ⓘ Symbols: $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient A&S Ref: 9.2.14 Referenced by: §10.18(iii) Permalink: http://dlmf.nist.gov/10.17.E12 Encodings: TeX, pMML, png See also: Annotations for §10.17(ii), §10.17 and Ch.10

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

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

For (10.17.5) and (10.17.6) write

10.17.13		$$\begin{array}{c}\hfill H_{\nu }^{(1)}\left(z\right)\hfill \\ \hfill H_{\nu }^{(2)}\left(z\right)\hfill \end{array}\}={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{\pm \mathrm{i}\omega }\left(\sum _{k=0}^{\mathrm{\ell }-1}{(\pm \mathrm{i})}^k\frac{a_k(\nu )}{z^k}+R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right),$$
$\mathrm{\ell }=1,2,\mathrm{\dots }$.
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient, $\omega$ and $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder Referenced by: §10.17(iv), §10.17(v) Permalink: http://dlmf.nist.gov/10.17.E13 Encodings: TeX, pMML, png See also: Annotations for §10.17(iv), §10.17 and Ch.10

Then

10.17.14		$$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right),$$
ⓘ Defines: $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally) Symbols: $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, ${𝒱}_{a,b}\left(f\right)$: 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 ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$. Reported 2014-09-27 by Gergő Nemes See also: Annotations for §10.17(iv), §10.17 and Ch.10

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

10.17.15
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable and $\chi (\mathrm{\ell })$ Referenced by: §10.17(iv) Permalink: http://dlmf.nist.gov/10.17.E15 Encodings: TeX, pMML, png See also: Annotations for §10.17(iv), §10.17 and Ch.10

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

As in §9.7(v) denote

10.17.16		$$G_p\left(z\right)=\frac{{\mathrm{e}}^z}{2\pi }\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }(1-p,z),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $G_p\left(z\right)$: rescaled terminant function and $z$: complex variable Referenced by: §10.40(iv) Permalink: http://dlmf.nist.gov/10.17.E16 Encodings: TeX, pMML, png See also: Annotations for §10.17(v), §10.17 and Ch.10

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

10.17.17		$$R_{\mathrm{\ell }}^{\pm }(\nu ,z)={(-1)}^{\mathrm{\ell }}2\cos \left(\nu \pi \right)\left(\sum _{k=0}^{m-1}{(\pm \mathrm{i})}^k\frac{a_k(\nu )}{z^k}{G}_{\mathrm{\ell }-k}\left(\mp 2\mathrm{i}z\right)+R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $G_p\left(z\right)$: rescaled terminant function, $m$: integer, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder Permalink: http://dlmf.nist.gov/10.17.E17 Encodings: TeX, pMML, png See also: Annotations for §10.17(v), §10.17 and Ch.10

where

10.17.18		$$R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)=O\left({\mathrm{e}}^{-2|z|}{z}^{-m}\right),$$
$|\mathrm{ph}\left(z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}\right)|\le \pi$.
ⓘ Defines: $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally) Symbols: $O\left(x\right)$: 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, $\mathrm{ph}$: 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).