10 Bessel Functions Bessel and Hankel Functions 10.19 Asymptotic Expansions for Large Order 10.21 Zeros

§10.20 Uniform Asymptotic Expansions for Large Order

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large order, uniform
Referenced by:: §10.41(v), §10.57, §2.8(iii)
Permalink:: http://dlmf.nist.gov/10.20
See also:: Annotations for Ch.10

§10.20(i) Real Variables
§10.20(ii) Complex Variables
§10.20(iii) Double Asymptotic Properties

§10.20(i) Real Variables

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Keywords:: Bessel functions, Hankel functions, derivatives, uniform asymptotic expansions for large order
Notes:: See Olver (1997b, pp. 419–425).
Referenced by:: §10.19(iii), §10.19(iii), §10.20(ii), §10.74(i), §10.74(ii), 3rd item
Permalink:: http://dlmf.nist.gov/10.20.i
See also:: Annotations for §10.20 and Ch.10

Define $\zeta=\zeta(z)$ to be the solution of the differential equation

10.20.1		$\left(\frac{\mathrm{d}\zeta}{\mathrm{d}z}\right)^{2}=\frac{1-z^{2}}{\zeta z^{2}}$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $\zeta(z)$ : solution Permalink: http://dlmf.nist.gov/10.20.E1 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

that is infinitely differentiable on the interval $0<z<\infty$ , including $z=1$ . Then

10.20.2		$\frac{2}{3}\zeta^{\frac{3}{2}}=\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\,\mathrm{d% }t=\ln\left(\frac{1+\sqrt{1-z^{2}}}{z}\right)-\sqrt{1-z^{2}},$
$0<z\leq 1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable and $\zeta(z)$ : solution A&S Ref: 9.3.38 Referenced by: §10.20(ii) Permalink: http://dlmf.nist.gov/10.20.E2 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

10.20.3		$\frac{2}{3}(-\zeta)^{\frac{3}{2}}=\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\,% \mathrm{d}t=\sqrt{z^{2}-1}-\operatorname{arcsec}z,$
$1\leq z<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcsec}\NVar{z}$ : arcsecant function, $z$ : complex variable and $\zeta(z)$ : solution A&S Ref: 9.3.39 Referenced by: §10.20(ii), §10.21(viii) Permalink: http://dlmf.nist.gov/10.20.E3 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

all functions taking their principal values, with $\zeta=\infty,0,-\infty$ , corresponding to $z=0,1,\infty$ , respectively.

As $\nu\to\infty$ through positive real values

10.20.4	$\displaystyle J_{\nu}\left(\nu z\right)$	$\displaystyle\sim\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}\*\left(% \frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}% \sum_{k=0}^{\infty}\frac{A_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{Ai}'\left% (\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B_{% k}(\zeta)}{\nu^{2k}}\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : Poincaré asymptotic expansion, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\zeta(z)$ : solution, $A_{k}(\zeta)$ : coefficients and $B_{k}(\zeta)$ : coefficients A&S Ref: 9.3.35 Referenced by: §10.20(ii), §10.20(iii) Permalink: http://dlmf.nist.gov/10.20.E4 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10
10.20.5	$\displaystyle Y_{\nu}\left(\nu z\right)$	$\displaystyle\sim-\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}}\left(\frac% {\operatorname{Bi}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_% {k=0}^{\infty}\frac{A_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{Bi}'\left(\nu^% {\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B_{k}(% \zeta)}{\nu^{2k}}\right),$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\zeta(z)$ : solution, $A_{k}(\zeta)$ : coefficients and $B_{k}(\zeta)$ : coefficients A&S Ref: 9.3.36 Permalink: http://dlmf.nist.gov/10.20.E5 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

10.20.6		$\rselection{{H^{(1)}_{\nu}}\left(\nu z\right)\\ {H^{(2)}_{\nu}}\left(\nu z\right)}\sim 2e^{\mp\pi i/3}\left(\frac{4\zeta}{1-z^% {2}}\right)^{\frac{1}{4}}\left(\frac{\operatorname{Ai}\left(e^{\pm 2\pi i/3}% \nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{k=0}^{\infty}\frac{A_{k% }(\zeta)}{\nu^{2k}}+\frac{e^{\pm 2\pi i/3}\operatorname{Ai}'\left(e^{\pm 2\pi i% /3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B% _{k}(\zeta)}{\nu^{2k}}\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\zeta(z)$ : solution, $A_{k}(\zeta)$ : coefficients and $B_{k}(\zeta)$ : coefficients A&S Ref: 9.3.37 Referenced by: §10.20(iii), §10.41(v) Permalink: http://dlmf.nist.gov/10.20.E6 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

10.20.7	$\displaystyle J_{\nu}'\left(\nu z\right)$	$\displaystyle\sim-\frac{2}{z}\left(\frac{1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}% \*\left(\frac{\operatorname{Ai}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac% {4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{% Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{2}{3}}}\sum_{k=0}^{\infty}% \frac{D_{k}(\zeta)}{\nu^{2k}}\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : Poincaré asymptotic expansion, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\zeta(z)$ : solution, $C_{k}(\zeta)$ : coefficients and $D_{k}(\zeta)$ : coefficients A&S Ref: 9.3.43 Permalink: http://dlmf.nist.gov/10.20.E7 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10
10.20.8	$\displaystyle Y_{\nu}'\left(\nu z\right)$	$\displaystyle\sim\frac{2}{z}\left(\frac{1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}% \*\left(\frac{\operatorname{Bi}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac% {4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{% Bi}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{2}{3}}}\sum_{k=0}^{\infty}% \frac{D_{k}(\zeta)}{\nu^{2k}}\right),$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\zeta(z)$ : solution, $C_{k}(\zeta)$ : coefficients and $D_{k}(\zeta)$ : coefficients A&S Ref: 9.3.44 Permalink: http://dlmf.nist.gov/10.20.E8 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

10.20.9		$\rselection{{H^{(1)}_{\nu}}'\left(\nu z\right)\\ {H^{(2)}_{\nu}}'\left(\nu z\right)}\sim\frac{4e^{\mp 2\pi i/3}}{z}\left(\frac{% 1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}\*\left(\frac{e^{\mp 2\pi i/3}% \operatorname{Ai}\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{% \operatorname{Ai}'\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{2}{3}}}\sum_{k=0}^{\infty}\frac{D_{k}(\zeta)}{\nu^{2k}}\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\zeta(z)$ : solution, $C_{k}(\zeta)$ : coefficients and $D_{k}(\zeta)$ : coefficients A&S Ref: 9.3.45 Referenced by: §10.20(ii) Permalink: http://dlmf.nist.gov/10.20.E9 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20 and Ch.10

uniformly for $z$ $\in(0,\infty)$ in all cases, where $\operatorname{Ai}$ and $\operatorname{Bi}$ are the Airy functions (§9.2).

In the following formulas for the coefficients $A_{k}(\zeta)$ , $B_{k}(\zeta)$ , $C_{k}(\zeta)$ , and $D_{k}(\zeta)$ , $u_{k}$ , $v_{k}$ are the constants defined in §9.7(i), and $U_{k}(p)$ , $V_{k}(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii).

Interval $0<z<1$

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See also:: Annotations for §10.20(i), §10.20 and Ch.10

10.20.10		$A_{k}(\zeta)=\sum_{j=0}^{2k}(\tfrac{3}{2})^{j}v_{j}\zeta^{-3j/2}U_{2k-j}\left(% (1-z^{2})^{-\frac{1}{2}}\right),$
ⓘ Defines: $A_{k}(\zeta)$ : coefficients (locally) Symbols: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : solution, $v_{k}$ : constants and $U_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.40 Referenced by: §10.20(i), §10.41(v), §10.74(i) Permalink: http://dlmf.nist.gov/10.20.E10 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

10.20.11		$B_{k}(\zeta)=-\zeta^{-\frac{1}{2}}\sum_{j=0}^{2k+1}(\tfrac{3}{2})^{j}u_{j}% \zeta^{-3j/2}U_{2k-j+1}\left((1-z^{2})^{-\frac{1}{2}}\right),$
ⓘ Defines: $B_{k}(\zeta)$ : coefficients (locally) Symbols: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : solution, $u_{k}$ : constants and $U_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.40 Referenced by: §10.21(viii), §10.41(v) Permalink: http://dlmf.nist.gov/10.20.E11 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

10.20.12		$C_{k}(\zeta)=-\zeta^{\frac{1}{2}}\sum_{j=0}^{2k+1}(\tfrac{3}{2})^{j}v_{j}\zeta% ^{-3j/2}V_{2k-j+1}\left((1-z^{2})^{-\frac{1}{2}}\right),$
ⓘ Defines: $C_{k}(\zeta)$ : coefficients (locally) Symbols: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : solution, $v_{k}$ : constants and $V_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.46 Referenced by: §10.21(viii) Permalink: http://dlmf.nist.gov/10.20.E12 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

10.20.13		$D_{k}(\zeta)=\sum_{j=0}^{2k}(\tfrac{3}{2})^{j}u_{j}\zeta^{-3j/2}V_{2k-j}\left(% (1-z^{2})^{-\frac{1}{2}}\right).$
ⓘ Defines: $D_{k}(\zeta)$ : coefficients (locally) Symbols: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : solution, $u_{k}$ : constants and $V_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.46 Referenced by: §10.20(i), §10.74(i) Permalink: http://dlmf.nist.gov/10.20.E13 Encodings: TeX, pMML, png See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

Interval $1<z<\infty$

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See also:: Annotations for §10.20(i), §10.20 and Ch.10

In formulas (10.20.10)–(10.20.13) replace $\zeta^{\frac{1}{2}}$ , $\zeta^{-\frac{1}{2}}$ , $\zeta^{-3j/2}$ , and $(1-z^{2})^{-\frac{1}{2}}$ by $-i(-\zeta)^{\frac{1}{2}}$ , $i(-\zeta)^{-\frac{1}{2}}$ , $i^{3j}(-\zeta)^{-3j/2}$ , and $i(z^{2}-1)^{-\frac{1}{2}}$ , respectively.

Note: Another way of arranging the above formulas for the coefficients $A_{k}(\zeta),B_{k}(\zeta),C_{k}(\zeta)$ , and $D_{k}(\zeta)$ would be by analogy with (12.10.42) and (12.10.46). In this way there is less usage of many-valued functions.

Values at $\zeta=0$

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See also:: Annotations for §10.20(i), §10.20 and Ch.10

10.20.14	$\displaystyle A_{0}(0)$	$\displaystyle=1,$
	$\displaystyle A_{1}(0)$	$\displaystyle=-\tfrac{1}{225},$
	$\displaystyle A_{2}(0)$	$\displaystyle=\tfrac{1\;51439}{2182\;95000},$
	$\displaystyle A_{3}(0)$	$\displaystyle=-\tfrac{8872\;78009}{250\;49351\;25000},\\$
	$\displaystyle B_{0}(0)$	$\displaystyle=\tfrac{1}{70}2^{\frac{1}{3}},$
	$\displaystyle B_{1}(0)$	$\displaystyle=-\tfrac{1213}{10\;23750}2^{\frac{1}{3}},$
	$\displaystyle B_{2}(0)$	$\displaystyle=\tfrac{1\;65425\;37833}{3774\;32055\;00000}2^{\frac{1}{3}},$
	$\displaystyle B_{3}(0)$	$\displaystyle=-\tfrac{959\;71711\;84603}{25\;47666\;37125\;00000}2^{\frac{1}{3% }}.$
ⓘ Symbols: $A_{k}(\zeta)$ : coefficients and $B_{k}(\zeta)$ : coefficients Referenced by: §10.20(i), Erratum (V1.0.5) for Equation (10.20.14) Permalink: http://dlmf.nist.gov/10.20.E14 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png Errata (effective with 1.0.5): Originally the value given for $B_{3}(0)$ was given incorrectly as $B_{3}(0)=-\tfrac{430\;99056\;39368\;59253}{5\;68167\;34399\;42500\;00000}2^{% \frac{1}{3}}$ . Reported 2012-05-11 by Antony Lee See also: Annotations for §10.20(i), §10.20(i), §10.20 and Ch.10

Each of the coefficients $A_{k}(\zeta)$ , $B_{k}(\zeta)$ , $C_{k}(\zeta)$ , and $D_{k}(\zeta)$ , $k=0,1,2,\dotsc$ , is real and infinitely differentiable on the interval $-\infty<\zeta<\infty$ . For (10.20.14) and further information on the coefficients see Temme (1997).

For numerical tables of $\zeta=\zeta(z)$ , $(4\zeta/(1-z^{2}))^{\frac{1}{4}}$ and $A_{k}(\zeta)$ , $B_{k}(\zeta)$ , $C_{k}(\zeta)$ , and $D_{k}(\zeta)$ see Olver (1962, pp. 28–42).

§10.20(ii) Complex Variables

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Keywords:: Bessel functions, asymptotic expansions for large order, resurgence properties of coefficients
Notes:: See Olver (1997b, pp. 419–425) and Olver (1954).
Referenced by:: §10.14, §10.74(i)
Permalink:: http://dlmf.nist.gov/10.20.ii
See also:: Annotations for §10.20 and Ch.10

The function $\zeta=\zeta(z)$ given by (10.20.2) and (10.20.3) can be continued analytically to the $z$ -plane cut along the negative real axis. Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2.

The equations of the curved boundaries $D_{1}E_{1}$ and $D_{2}E_{2}$ in the $\zeta$ -plane are given parametrically by

10.20.15		$\zeta=(\tfrac{3}{2})^{\frac{2}{3}}(\tau\mp i\pi)^{\frac{2}{3}},$
$0\leq\tau<\infty$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit and $\zeta(z)$ : solution Permalink: http://dlmf.nist.gov/10.20.E15 Encodings: TeX, pMML, png See also: Annotations for §10.20(ii), §10.20 and Ch.10

respectively.

The curves $BP_{1}E_{1}$ and $BP_{2}E_{2}$ in the $z$ -plane are the inverse maps of the line segments

10.20.16		$\zeta=e^{\mp i\pi/3}\tau,$
$0\leq\tau\leq(\tfrac{3}{2}\pi)^{\frac{2}{3}}$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $\zeta(z)$ : solution Permalink: http://dlmf.nist.gov/10.20.E16 Encodings: TeX, pMML, png See also: Annotations for §10.20(ii), §10.20 and Ch.10

respectively. They are given parametrically by

10.20.17		$z=\pm(\tau\coth\tau-\tau^{2})^{\frac{1}{2}}\pm\mathrm{i}(\tau^{2}-\tau\tanh% \tau)^{\frac{1}{2}},$
$0\leq\tau\leq\tau_{0}$ ,
ⓘ Symbols: $\coth\NVar{z}$ : hyperbolic cotangent function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\mathrm{i}$ : imaginary unit and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.20.E17 Encodings: TeX, pMML, png See also: Annotations for §10.20(ii), §10.20 and Ch.10

where $\tau_{0}=1.19968\ldots$ is the positive root of the equation $\tau=\coth\tau$ . The points $P_{1},P_{2}$ where these curves intersect the imaginary axis are $\pm ic$ , where

10.20.18		$c=(\tau_{0}^{2}-1)^{\frac{1}{2}}=0.66274\dotsc.$
ⓘ Defines: $c$ (locally) Referenced by: §10.41(iii) Permalink: http://dlmf.nist.gov/10.20.E18 Encodings: TeX, pMML, png See also: Annotations for §10.20(ii), §10.20 and Ch.10

The eye-shaped closed domain in the uncut $z$ -plane that is bounded by $BP_{1}E_{1}$ and $BP_{2}E_{2}$ is denoted by $\mathbf{K}$ ; see Figure 10.20.3.

See accompanying text — Figure 10.20.1: $z$ -plane. $P_{1}$ and $P_{2}$ are the points $\pm ic$ . $c=0.66274\dotsc$ . Magnify

As $\nu\to\infty$ through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for $|\operatorname{ph}z|\leq\pi-\delta$ , the coefficients $A_{k}(\zeta)$ , $B_{k}(\zeta)$ , $C_{k}(\zeta)$ , and $D_{k}(\zeta)$ , being the analytic continuations of the functions defined in §10.20(i) when $\zeta$ is real.

For proofs of the above results and for error bounds and extensions of the regions of validity see Olver (1997b, pp. 419–425). For extensions to complex $\nu$ see Olver (1954). For resurgence properties of the coefficients (§2.7(ii)) see Howls and Olde Daalhuis (1999). For further results see Dunster (2001a), Wang and Wong (2002), and Paris (2004).