10 Bessel Functions Bessel and Hankel Functions 10.16 Relations to Other Functions 10.18 Modulus and Phase Functions

§10.17 Asymptotic Expansions for Large Argument

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Keywords:: Bessel functions, asymptotic expansions for large argument
Permalink:: http://dlmf.nist.gov/10.17
See also:: Annotations for Ch.10

§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

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Keywords:: Hankel functions, Hankel’s expansions, asymptotic expansions for large argument, for Bessel and Hankel functions
Notes:: See Olver (1997b, pp. 237–242).
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
See also:: Annotations for §10.17 and Ch.10

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

10.17.1 $a_{k}(\nu)=\frac{(4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k-1)^{2})}{% k!8^{k}}=\frac{{\left(\frac{1}{2}-\nu\right)_{k}}{\left(\frac{1}{2}+\nu\right)% _{k}}}{(-2)^{k}k!},$ $k\geq 1$ ,

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Defines:: $a_{k}(\nu)$ : polynomial coefficient (locally)
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), : 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 $\omega=z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi,$

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Defines:: $\omega$ (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : 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 $J_{\nu}\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\*\left(% \cos\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{a_{2k}(\nu)}{z^{2k}}-\sin\omega\sum% _{k=0}^{\infty}(-1)^{k}\frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\pi-\delta$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, : nonnegative integer, : 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 $Y_{\nu}\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\*\left(% \sin\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{a_{2k}(\nu)}{z^{2k}}+\cos\omega\sum% _{k=0}^{\infty}(-1)^{k}\frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\pi-\delta$ ,

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Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, : nonnegative integer, : 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 ${H^{(1)}_{\nu}}\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{% i\omega}\sum_{k=0}^{\infty}i^{k}\frac{a_{k}(\nu)}{z^{k}},$ $-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta$ ,

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Symbols:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : phase, : nonnegative integer, : 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 ${H^{(2)}_{\nu}}\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}e^{% -i\omega}\sum_{k=0}^{\infty}(-i)^{k}\frac{a_{k}(\nu)}{z^{k}},$ $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$ ,

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Symbols:: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : phase, : nonnegative integer, : 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 $z^{\frac{1}{2}}=\exp\left(\tfrac{1}{2}\ln|z|+\tfrac{1}{2}i\operatorname{ph}z% \right).$

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Symbols:: $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase and : 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 $\operatorname{ph}z$ 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

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large argument, derivatives
Notes:: These results follow by differentiation of the corresponding expansions in §10.17(i); compare §2.1(iii).
Referenced by:: §10.40(i), §10.67(i)
Permalink:: http://dlmf.nist.gov/10.17.ii
See also:: Annotations for §10.17 and Ch.10

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 $b_{k}(\nu)=\frac{\left((4\nu^{2}-1^{2})(4\nu^{2}-3^{2})\cdots(4\nu^{2}-(2k-3)^% {2})\right)(4\nu^{2}+4k^{2}-1)}{k!8^{k}}.$

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Defines:: $b_{k}(\nu)$ : polynomial coefficient (locally)
Symbols:: : factorial (as in ), : 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 $J_{\nu}'\left(z\right)\sim-\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\left(% \sin\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{b_{2k}(\nu)}{z^{2k}}+\cos\omega\sum% _{k=0}^{\infty}(-1)^{k}\frac{b_{2k+1}(\nu)}{z^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\pi-\delta$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, : nonnegative integer, : 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 $Y_{\nu}'\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\left(\cos% \omega\sum_{k=0}^{\infty}(-1)^{k}\frac{b_{2k}(\nu)}{z^{2k}}-\sin\omega\sum_{k=% 0}^{\infty}(-1)^{k}\frac{b_{2k+1}(\nu)}{z^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\pi-\delta$ ,

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Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, : nonnegative integer, : 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 ${H^{(1)}_{\nu}}'\left(z\right)\sim i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% e^{i\omega}\sum_{k=0}^{\infty}i^{k}\frac{b_{k}(\nu)}{z^{k}},$ $-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta$ ,

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Symbols:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : phase, : nonnegative integer, : 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 ${H^{(2)}_{\nu}}'\left(z\right)\sim-i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% e^{-i\omega}\sum_{k=0}^{\infty}(-i)^{k}\frac{b_{k}(\nu)}{z^{k}},$ $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$ .

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Symbols:: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $\operatorname{ph}$ : phase, : nonnegative integer, : 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

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

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large argument, error bounds
Notes:: See Watson (1944, pp. 205–206) and Olver (1997b, pp. 268–269).
Permalink:: http://dlmf.nist.gov/10.17.iii
See also:: Annotations for §10.17 and Ch.10

In the expansions (10.17.3) and (10.17.4) assume that $\nu\geq 0$ and . 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 . 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

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Notes:: See Olver (1997b, pp. 266–267).
Referenced by:: §10.49(i), §10.74(i)
Permalink:: http://dlmf.nist.gov/10.17.iv
See also:: Annotations for §10.17 and Ch.10

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 $\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm i% \infty}\left(t^{-\ell}\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,% \pm i\infty}\left(t^{-1}\right)\right),$

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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, : 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 $\mathcal{V}_{z,i\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&0% \leq\operatorname{ph}z\leq\pi,\\ \chi(\ell)|z|^{-\ell},&\parbox[t]{224.03743pt}{$-\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]{224.03743pt}{$-\pi<\operatorname{ph}z% \leq-\tfrac{1}{2}\pi$ or $\tfrac{3}{2}\pi\leq\operatorname{ph}z<2\pi$,}\end{cases}$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation, : complex variable and $\chi(\ell)$
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 $\chi(\ell)=\pi^{\frac{1}{2}}\Gamma\left(\tfrac{1}{2}\ell+1\right)/\Gamma\left(% \tfrac{1}{2}\ell+\tfrac{1}{2}\right)$ ; 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

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large argument, error bounds, exponentially-improved
Notes:: See Olver (1991b, Theorem 1) or Olver (1993a, Theorem 1.1), and (10.16.6).
Referenced by:: §10.74(i)
Permalink:: http://dlmf.nist.gov/10.17.v
See also:: Annotations for §10.17 and Ch.10

As in §9.7(v) denote

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $G_{\NVar{p}}\left(\NVar{z}\right)$ : rescaled terminant function and : 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 $\Gamma\left(1-p,z\right)$ is the incomplete gamma function (§8.2(i)). Then in (10.17.13) as $z\to\infty$ with $|\ell-2|z||$ bounded and ( $\geq 0$ ) fixed,

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

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