§10.17 Asymptotic Expansions for Large Argument

Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.17

§10.17(i) Hankel’s Expansions
§10.17(ii) Asymptotic Expansions of Derivatives
§10.17(iii) Error Bounds for Real Argument and Order
§10.17(iv) Error Bounds for Complex Argument and Order
§10.17(v) Exponentially-Improved Expansions

§10.17(i) Hankel’s Expansions

Notes:: See Olver (1997b, pp. 237–242).
Keywords:: Hankel functions, Hankel’s expansions
Referenced by:: §10.17(ii), §10.17(ii), §10.40(i), §10.67(i), §10.74(i), §10.74(i), §10.74(ii)
Permalink:: http://dlmf.nist.gov/10.17.i

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
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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
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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 $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}e^{{i\omega}}\sum _{{k=0}}^{\infty}i^{k}\frac{a_{k}(\nu)}{z^{k}},$ $-\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq 2\pi-\delta$ ,

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $a_{k}(\nu)$ : expansion 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
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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 $z^{{\frac{1}{2}}}=\mathop{\exp\/}\nolimits\left(\tfrac{1}{2}\mathop{\ln\/}\nolimits|z|+\tfrac{1}{2}i\mathop{\mathrm{ph}\/}\nolimits z\right).$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.17.E7
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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

Notes:: These results follow by differentiation of the corresponding expansions in §10.17(i); compare §2.1(iii).
Keywords:: Bessel functions, Hankel functions, derivatives
Referenced by:: §10.40(i), §10.67(i)
Permalink:: http://dlmf.nist.gov/10.17.ii

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

10.17.9 ${\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim-\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\left(\mathop{\sin\/}\nolimits\omega\sum _{{k=0}}^{\infty}(-1)^{k}\frac{b_{{2k}}(\nu)}{z^{{2k}}}+\mathop{\cos\/}\nolimits\omega\sum _{{k=0}}^{\infty}(-1)^{k}\frac{b_{{2k+1}}(\nu)}{z^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $\omega$ and $b_{k}(\nu)$ : expansion
A&S Ref:: 9.2.11
Permalink:: http://dlmf.nist.gov/10.17.E9
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10.17.10 ${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\left(\mathop{\cos\/}\nolimits\omega\sum _{{k=0}}^{\infty}(-1)^{k}\frac{b_{{2k}}(\nu)}{z^{{2k}}}-\mathop{\sin\/}\nolimits\omega\sum _{{k=0}}^{\infty}(-1)^{k}\frac{b_{{2k+1}}(\nu)}{z^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ,

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $\omega$ and $b_{k}(\nu)$ : expansion
A&S Ref:: 9.2.12
Permalink:: http://dlmf.nist.gov/10.17.E10
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10.17.11 ${\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim i\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}e^{{i\omega}}\sum _{{k=0}}^{\infty}i^{k}\frac{b_{k}(\nu)}{z^{k}},$ $-\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq 2\pi-\delta$ ,

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $\omega$ and $b_{k}(\nu)$ : expansion
A&S Ref:: 9.2.13
Referenced by:: §10.18(iii)
Permalink:: http://dlmf.nist.gov/10.17.E11
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10.17.12 ${\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim-i\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}e^{{-i\omega}}\sum _{{k=0}}^{\infty}(-i)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $-2\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi-\delta$ .

Symbols:: $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $\omega$ and $b_{k}(\nu)$ : expansion
A&S Ref:: 9.2.14
Referenced by:: §10.18(iii)
Permalink:: http://dlmf.nist.gov/10.17.E12
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§10.17(iii) Error Bounds for Real Argument and Order

Notes:: See Watson (1944, pp. 205–206) and Olver (1997b, pp. 268–269).
Keywords:: Bessel functions, Hankel functions
Permalink:: http://dlmf.nist.gov/10.17.iii

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

Notes:: See Olver (1997b, pp. 266–267).
Referenced by:: §10.49(i), §10.74(i)
Permalink:: http://dlmf.nist.gov/10.17.iv

For (10.17.5) and (10.17.6) write

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

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $e$ : base of exponential function, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $a_{k}(\nu)$ : expansion, $\omega$ and $R_{\ell}^{\pm}(\nu,z)$ : remainder
Referenced by:: §10.17(iv), §10.17(v)
Permalink:: http://dlmf.nist.gov/10.17.E13
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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
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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}$

Symbols:: $\imagpart{}$ : imaginary part, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathcal{V}_{{a,b}}\/}\nolimits\!\left(f\right)$ : total variation, $z$ : complex variable and $\chi(\ell)$
Referenced by:: §10.17(iv)
Permalink:: http://dlmf.nist.gov/10.17.E15
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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

Notes:: See Olver (1991b, Theorem 1) or Olver (1993a, Theorem 1.1), and (10.16.6).
Keywords:: Bessel functions, Hankel functions
Referenced by:: §10.74(i)
Permalink:: http://dlmf.nist.gov/10.17.v

As in §9.7(v) denote

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{G_{{p}}\/}\nolimits\!\left(z\right)$ : product of gamma and incomplete gamma functions and $z$ : complex variable
Referenced by:: §10.40(iv)
Permalink:: http://dlmf.nist.gov/10.17.E16
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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 $R_{{\ell}}^{{\pm}}(\nu,z)=(-1)^{{\ell}}2\mathop{\cos\/}\nolimits(\nu\pi)\*\left(\sum _{{k=0}}^{{m-1}}(\pm i)^{k}\frac{a_{k}(\nu)}{z^{k}}\mathop{G_{{\ell-k}}\/}\nolimits\!\left(\mp 2iz\right)+R_{{m,\ell}}^{{\pm}}(\nu,z)\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{G_{{p}}\/}\nolimits\!\left(z\right)$ : product of gamma and incomplete gamma functions, $m$ : integer, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $a_{k}(\nu)$ : expansion and $R_{\ell}^{\pm}(\nu,z)$ : remainder
Permalink:: http://dlmf.nist.gov/10.17.E17
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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).