10 Bessel Functions Bessel and Hankel Functions 10.18 Modulus and Phase Functions 10.20 Uniform Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large order
Permalink:: http://dlmf.nist.gov/10.19
See also:: Annotations for Ch.10

§10.19(i) Asymptotic Forms

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large order, asymptotic forms, derivatives
Notes:: These results follow from (10.2.2), (10.2.3), (10.4.3), (10.8.1), (5.5.3), (5.11.3).
Referenced by:: §3.6(vi)
Permalink:: http://dlmf.nist.gov/10.19.i
See also:: Annotations for §10.19 and Ch.10

If $\nu\to\infty$ through positive real values, with $z$ $(\neq 0)$ fixed, then

10.19.1		$J_{\nu}\left(z\right)\sim\frac{1}{\sqrt{2\pi\nu}}\left(\frac{ez}{2\nu}\right)^% {\nu},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.3.1 Referenced by: §10.41(i), §33.9(i) Permalink: http://dlmf.nist.gov/10.19.E1 Encodings: TeX, pMML, png See also: Annotations for §10.19(i), §10.19 and Ch.10

10.19.2		$Y_{\nu}\left(z\right)\sim-i{H^{(1)}_{\nu}}\left(z\right)\sim i{H^{(2)}_{\nu}}% \left(z\right)\sim-\sqrt{\frac{2}{\pi\nu}}\left(\frac{ez}{2\nu}\right)^{-\nu}.$
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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), $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.3.1 Referenced by: §10.41(i) Permalink: http://dlmf.nist.gov/10.19.E2 Encodings: TeX, pMML, png See also: Annotations for §10.19(i), §10.19 and Ch.10

§10.19(ii) Debye’s Expansions

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Keywords:: Bessel functions, Debye’s expansions, asymptotic expansions for large order, derivatives
Notes:: For (10.19.3) and (10.19.6) see Watson (1944, pp. 241–245) and Bickley et al. (1952, p. xxxv). The expansions for the derivatives are established in a similar manner, with the coefficients calculated by term-by-term differentiation; compare §2.1(iii).
Referenced by:: §11.11(iii), Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/10.19.ii
Errata (effective with 1.0.7):: The last sentence of this subsection, referring to Olver (1997b, p. 382) for (10.19.6), has been corrected. Previously it referred to Olver (1997b, p. 382) for (10.19.3).
Reported 2013-08-30 by Gergő Nemes
See also:: Annotations for §10.19 and Ch.10

If $\nu\to\infty$ through positive real values with $\alpha$ $(>0)$ fixed, then

10.19.3	$\displaystyle J_{\nu}\left(\nu\operatorname{sech}\alpha\right)$	$\displaystyle\sim\frac{e^{\nu(\tanh\alpha-\alpha)}}{(2\pi\nu\tanh\alpha)^{% \frac{1}{2}}}\sum_{k=0}^{\infty}\frac{U_{k}(\coth\alpha)}{\nu^{k}},$
10.19.3	$\displaystyle Y_{\nu}\left(\nu\operatorname{sech}\alpha\right)$	$\displaystyle\sim-\frac{e^{\nu(\alpha-\tanh\alpha)}}{(\tfrac{1}{2}\pi\nu\tanh% \alpha)^{\frac{1}{2}}}\*\sum_{k=0}^{\infty}(-1)^{k}\frac{U_{k}(\coth\alpha)}{% \nu^{k}},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\coth\NVar{z}$ : hyperbolic cotangent function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\tanh\NVar{z}$ : hyperbolic tangent function, $k$ : nonnegative integer, $\nu$ : complex parameter and $U_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.7, 9.3.8 Referenced by: §10.19(ii) Permalink: http://dlmf.nist.gov/10.19.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.19(ii), §10.19 and Ch.10

10.19.4	$\displaystyle J_{\nu}'\left(\nu\operatorname{sech}\alpha\right)$	$\displaystyle\sim\left(\frac{\sinh\left(2\alpha\right)}{4\pi\nu}\right)^{\frac% {1}{2}}e^{\nu(\tanh\alpha-\alpha)}\sum_{k=0}^{\infty}\frac{V_{k}(\coth\alpha)}% {\nu^{k}},$
10.19.4	$\displaystyle Y_{\nu}'\left(\nu\operatorname{sech}\alpha\right)$	$\displaystyle\sim\left(\frac{\sinh\left(2\alpha\right)}{\pi\nu}\right)^{\frac{% 1}{2}}e^{\nu(\alpha-\tanh\alpha)}\sum_{k=0}^{\infty}(-1)^{k}\frac{V_{k}(\coth% \alpha)}{\nu^{k}}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\coth\NVar{z}$ : hyperbolic cotangent function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $k$ : nonnegative integer, $\nu$ : complex parameter and $V_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.11, 9.3.12 Permalink: http://dlmf.nist.gov/10.19.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.19(ii), §10.19 and Ch.10

If $\nu\to\infty$ through positive real values with $\beta$ $\left(\in\left(0,\tfrac{1}{2}\pi\right)\right)$ fixed, and

10.19.5		$\xi=\nu(\tan\beta-\beta)-\tfrac{1}{4}\pi,$
ⓘ Defines: $\xi$ (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\tan\NVar{z}$ : tangent function, $\nu$ : complex parameter and $\beta$ Permalink: http://dlmf.nist.gov/10.19.E5 Encodings: TeX, pMML, png See also: Annotations for §10.19(ii), §10.19 and Ch.10

then

10.19.6	$\displaystyle J_{\nu}\left(\nu\sec\beta\right)$	$\displaystyle\sim\left(\frac{2}{\pi\nu\tan\beta}\right)^{\frac{1}{2}}\*\left(% \cos\xi\sum_{k=0}^{\infty}\frac{U_{2k}(i\cot\beta)}{\nu^{2k}}-i\sin\xi\sum_{k=% 0}^{\infty}\frac{U_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$
10.19.6	$\displaystyle Y_{\nu}\left(\nu\sec\beta\right)$	$\displaystyle\sim\left(\frac{2}{\pi\nu\tan\beta}\right)^{\frac{1}{2}}\*\left(% \sin\xi\sum_{k=0}^{\infty}\frac{U_{2k}(i\cot\beta)}{\nu^{2k}}+i\cos\xi\sum_{k=% 0}^{\infty}\frac{U_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $\cot\NVar{z}$ : cotangent function, $\mathrm{i}$ : imaginary unit, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $k$ : nonnegative integer, $\nu$ : complex parameter, $\beta$ , $\xi$ and $U_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.15--9.3.18 Referenced by: §10.19(ii), §10.19(ii), §11.6(iii) Permalink: http://dlmf.nist.gov/10.19.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.19(ii), §10.19 and Ch.10

10.19.7	$\displaystyle J_{\nu}'\left(\nu\sec\beta\right)$	$\displaystyle\sim\left(\frac{\sin\left(2\beta\right)}{\pi\nu}\right)^{\frac{1}% {2}}\*\left(-\sin\xi\sum_{k=0}^{\infty}\frac{V_{2k}(i\cot\beta)}{\nu^{2k}}-i% \cos\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$
10.19.7	$\displaystyle Y_{\nu}'\left(\nu\sec\beta\right)$	$\displaystyle\sim\left(\frac{\sin\left(2\beta\right)}{\pi\nu}\right)^{\frac{1}% {2}}\*\left(\cos\xi\sum_{k=0}^{\infty}\frac{V_{2k}(i\cot\beta)}{\nu^{2k}}-i% \sin\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $\cot\NVar{z}$ : cotangent function, $\mathrm{i}$ : imaginary unit, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $\nu$ : complex parameter, $\beta$ , $\xi$ and $V_{k}(p)$ : polynomial coefficient A&S Ref: 9.3.19--9.3.22 Permalink: http://dlmf.nist.gov/10.19.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.19(ii), §10.19 and Ch.10

In these expansions $U_{k}(p)$ and $V_{k}(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii).

For error bounds for the first of (10.19.6) see Olver (1997b, p. 382).

§10.19(iii) Transition Region

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Keywords:: Bessel functions, Hankel functions, asymptotic expansions for large order, derivatives, transition region
Referenced by:: Erratum (V1.0.5) for References, Erratum (V1.0.7) for Usability
Permalink:: http://dlmf.nist.gov/10.19.iii
Addition (effective with 1.0.7):: The cross-reference to §10.20(i) has been added at the end of this subsection.
Addition (effective with 1.0.5):: The reference to Jentschura and Lötstedt (2012) has been added at the end of this subsection.
See also:: Annotations for §10.19 and Ch.10

As $\nu\to\infty$ , with $a(\in\mathbb{C})$ fixed,

10.19.8		$\displaystyle J_{\nu}\left(\nu+a\nu^{\frac{1}{3}}\right)$	$\displaystyle\sim\frac{2^{\frac{1}{3}}}{\nu^{\frac{1}{3}}}\operatorname{Ai}% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{P_{k}(a)}{\nu^{2k/3}}+% \frac{2^{\frac{2}{3}}}{\nu}\operatorname{Ai}'\left(-2^{\frac{1}{3}}a\right)% \sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$
	$\|\operatorname{ph}\nu\|\leq\tfrac{1}{2}\pi-\delta$ ,
		$\displaystyle Y_{\nu}\left(\nu+a\nu^{\frac{1}{3}}\right)$	$\displaystyle\sim-\frac{2^{\frac{1}{3}}}{\nu^{\frac{1}{3}}}\operatorname{Bi}% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{P_{k}(a)}{\nu^{2k/3}}-% \frac{2^{\frac{2}{3}}}{\nu}\operatorname{Bi}'\left(-2^{\frac{1}{3}}a\right)% \sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$
	$\|\operatorname{ph}\nu\|\leq\tfrac{1}{2}\pi-\delta$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $\nu$ : complex parameter, $\delta$ : small positive constant, $P_{k}(a)$ : polynomial coefficient and $Q_{k}(a)$ : polynomial coefficient A&S Ref: 9.3.23, 9.3.24 Referenced by: §10.19(iii) Permalink: http://dlmf.nist.gov/10.19.E8 Encodings: TeX, TeX, pMML, pMML, png, png Correction (effective with 1.0.27): Originally the polynomials $P_{k}(a)$ , $Q_{k}(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively. See also: Annotations for §10.19(iii), §10.19 and Ch.10

Also,

10.19.9		$\rselection{{H^{(1)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim\frac{2^{\frac{4}{3}}}{% \nu^{\frac{1}{3}}}e^{\mp\pi i/3}\operatorname{Ai}\left(e^{\mp\pi i/3}2^{\frac{% 1}{3}}a\right)\sum_{k=0}^{\infty}\frac{P_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{5}% {3}}}{\nu}e^{\pm\pi i/3}\operatorname{Ai}'\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a% \right)\sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$
ⓘ 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, $\nu$ : complex parameter, $P_{k}(a)$ : polynomial coefficient and $Q_{k}(a)$ : polynomial coefficient Permalink: http://dlmf.nist.gov/10.19.E9 Encodings: TeX, pMML, png Correction (effective with 1.0.27): Originally the polynomials $P_{k}(a)$ , $Q_{k}(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively. See also: Annotations for §10.19(iii), §10.19 and Ch.10

with sectors of validity $-\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}\nu\leq\tfrac{3}{2}\pi-\delta$ . Here $\operatorname{Ai}$ and $\operatorname{Bi}$ are the Airy functions (§9.2), and

10.19.10	$\displaystyle P_{0}(a)$	$\displaystyle=1,$
	$\displaystyle P_{1}(a)$	$\displaystyle=-\tfrac{1}{5}a,$
	$\displaystyle P_{2}(a)$	$\displaystyle=-\tfrac{9}{100}a^{5}+\tfrac{3}{35}a^{2},$
	$\displaystyle P_{3}(a)$	$\displaystyle=\tfrac{957}{7000}a^{6}-\tfrac{173}{3150}a^{3}-\tfrac{1}{225},$
	$\displaystyle P_{4}(a)$	$\displaystyle=\tfrac{27}{20000}a^{10}-\tfrac{23573}{1\;47000}a^{7}+\tfrac{5903% }{1\;38600}a^{4}+\tfrac{947}{3\;46500}a,$
ⓘ Defines: $P_{k}(a)$ : polynomial coefficient (locally) Symbols: $k$ : nonnegative integer A&S Ref: 9.3.25 Permalink: http://dlmf.nist.gov/10.19.E10 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png Correction (effective with 1.0.27): Originally the polynomials $P_{k}(a)$ , were incorrectly described as Legendre polynomials. See also: Annotations for §10.19(iii), §10.19 and Ch.10

10.19.11	$\displaystyle Q_{0}(a)$	$\displaystyle=\tfrac{3}{10}a^{2},$
	$\displaystyle Q_{1}(a)$	$\displaystyle=-\tfrac{17}{70}a^{3}+\tfrac{1}{70},$
	$\displaystyle Q_{2}(a)$	$\displaystyle=-\tfrac{9}{1000}a^{7}+\tfrac{611}{3150}a^{4}-\tfrac{37}{3150}a,$
	$\displaystyle Q_{3}(a)$	$\displaystyle=\tfrac{549}{28000}a^{8}-\tfrac{1\;10767}{6\;93000}a^{5}+\tfrac{7% 9}{12375}a^{2}.$
ⓘ Defines: $Q_{k}(a)$ : polynomial coefficient (locally) Symbols: $k$ : nonnegative integer A&S Ref: 9.3.26 Referenced by: Erratum (V1.0.10) for Equation (10.19.11) Permalink: http://dlmf.nist.gov/10.19.E11 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png Correction (effective with 1.0.27): Originally the polynomials $Q_{k}(a)$ , were incorrectly described as integer-degree, zero-order associated Legendre functions of the second kind. Errata (effective with 1.0.10): Originally the first term on the right-hand side of the equation for $Q_{3}(a)$ was written incorrectly as $-\tfrac{549}{28000}a^{8}$ . Reported 2015-03-16 by Svante Janson See also: Annotations for §10.19(iii), §10.19 and Ch.10

10.19.12		$\displaystyle J_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)$	$\displaystyle\sim-\frac{2^{\frac{2}{3}}}{\nu^{\frac{2}{3}}}\operatorname{Ai}'% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}+% \frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}\operatorname{Ai}\left(-2^{\frac{1}{3% }}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$
	$\|\operatorname{ph}\nu\|\leq\tfrac{1}{2}\pi-\delta$ ,
		$\displaystyle Y_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)$	$\displaystyle\sim\frac{2^{\frac{2}{3}}}{\nu^{\frac{2}{3}}}\operatorname{Bi}'% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}-% \frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}\operatorname{Bi}\left(-2^{\frac{1}{3% }}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$
	$\|\operatorname{ph}\nu\|\leq\tfrac{1}{2}\pi-\delta$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $\nu$ : complex parameter, $\delta$ : small positive constant, $R_{k}(a)$ : polynomial coefficient and $S_{k}(a)$ : polynomial coefficient A&S Ref: 9.3.27, 9.3.28 Permalink: http://dlmf.nist.gov/10.19.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.19(iii), §10.19 and Ch.10

10.19.13		$\rselection{{H^{(1)}_{\nu}}'\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}'\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim-\frac{2^{\frac{5}{3}}% }{\nu^{\frac{2}{3}}}e^{\pm\pi i/3}\operatorname{Ai}'\left(e^{\mp\pi i/3}2^{% \frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}+\frac{2^{% \frac{4}{3}}}{\nu^{\frac{4}{3}}}e^{\mp\pi i/3}\operatorname{Ai}\left(e^{\mp\pi i% /3}2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$
ⓘ 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, $\nu$ : complex parameter, $R_{k}(a)$ : polynomial coefficient and $S_{k}(a)$ : polynomial coefficient Permalink: http://dlmf.nist.gov/10.19.E13 Encodings: TeX, pMML, png See also: Annotations for §10.19(iii), §10.19 and Ch.10

with sectors of validity $-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}\nu\leq\tfrac{3}{2}\pi-\delta$ and $-\tfrac{3}{2}\pi+\delta\leq\operatorname{ph}\nu\leq\tfrac{1}{2}\pi-\delta$ , respectively. Here

10.19.14	$\displaystyle R_{0}(a)$	$\displaystyle=1,$
	$\displaystyle R_{1}(a)$	$\displaystyle=-\tfrac{4}{5}a,$
	$\displaystyle R_{2}(a)$	$\displaystyle=-\tfrac{9}{100}a^{5}+\tfrac{57}{70}a^{2},$
	$\displaystyle R_{3}(a)$	$\displaystyle=\tfrac{699}{3500}a^{6}-\tfrac{2617}{3150}a^{3}+\tfrac{23}{3150},$
	$\displaystyle R_{4}(a)$	$\displaystyle=\tfrac{27}{20000}a^{10}-\tfrac{46631}{1\;47000}a^{7}+\tfrac{3889% }{4620}a^{4}-\tfrac{1159}{1\;15500}a,$
ⓘ Defines: $R_{k}(a)$ : polynomial coefficient (locally) Symbols: $k$ : nonnegative integer A&S Ref: 9.3.29 Permalink: http://dlmf.nist.gov/10.19.E14 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §10.19(iii), §10.19 and Ch.10

10.19.15	$\displaystyle S_{0}(a)$	$\displaystyle=\tfrac{3}{5}a^{3}-\tfrac{1}{5},$
	$\displaystyle S_{1}(a)$	$\displaystyle=-\tfrac{131}{140}a^{4}+\tfrac{1}{5}a,$
	$\displaystyle S_{2}(a)$	$\displaystyle=-\tfrac{9}{500}a^{8}+\tfrac{5437}{4500}a^{5}-\tfrac{593}{3150}a^% {2},$
	$\displaystyle S_{3}(a)$	$\displaystyle=\tfrac{369}{7000}a^{9}-\tfrac{9\;99443}{6\;93000}a^{6}+\tfrac{31% 727}{1\;73250}a^{3}+\tfrac{947}{3\;46500}.$
ⓘ Defines: $S_{k}(a)$ : polynomial coefficient (locally) Symbols: $k$ : nonnegative integer A&S Ref: 9.3.30 Permalink: http://dlmf.nist.gov/10.19.E15 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.19(iii), §10.19 and Ch.10

For proofs and also for the corresponding expansions for second derivatives see Olver (1952).

For higher coefficients in (10.19.8) in the case $a=0$ (that is, in the expansions of $J_{\nu}\left(\nu\right)$ and $Y_{\nu}\left(\nu\right)$ ), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). The last reference also includes the corresponding expansions for $J_{\nu}'\left(\nu\right)$ and $Y_{\nu}'\left(\nu\right)$ .

§10.19 Asymptotic Expansions for Large Order

Contents

§10.19(i) Asymptotic Forms

§10.19(ii) Debye’s Expansions

§10.19(iii) Transition Region