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

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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).

§10.20(iii) Double Asymptotic Properties

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large order, double asymptotic properties, uniform
Permalink:: http://dlmf.nist.gov/10.20.iii
See also:: Annotations for §10.20 and Ch.10

For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).

10.19 Asymptotic Expansions for Large Order 10.21 Zeros

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§10.20 Uniform Asymptotic Expansions for Large Order

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