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
§10.19(ii) Debye’s Expansions
§10.19(iii) Transition Region

§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 $(\neq 0)$ fixed, then

10.19.1 $J_{\nu}\left(z\right)\sim\frac{1}{\sqrt{2\pi\nu}}\left(\frac{ez}{2\nu}\right)^% {\nu},$

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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, : 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 $Y_{\nu}\left(z\right)\sim-i{H^{(1)}_{\nu}}\left(z\right)\sim i{H^{(2)}_{\nu}}% \left(z\right)\sim-\sqrt{\frac{2}{\pi\nu}}\left(\frac{ez}{2\nu}\right)^{-\nu}.$

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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, : 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 $\alpha$ fixed, then

10.19.3

$J_{\nu}\left(\nu\operatorname{sech}\alpha\right)\sim\frac{e^{\nu(\tanh\alpha-% \alpha)}}{(2\pi\nu\tanh\alpha)^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{U_{k}(% \coth\alpha)}{\nu^{k}},$

$Y_{\nu}\left(\nu\operatorname{sech}\alpha\right)\sim-\frac{e^{\nu(\alpha-\tanh% \alpha)}}{(\tfrac{1}{2}\pi\nu\tanh\alpha)^{\frac{1}{2}}}\*\sum_{k=0}^{\infty}(% -1)^{k}\frac{U_{k}(\coth\alpha)}{\nu^{k}},$

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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, : 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

$J_{\nu}'\left(\nu\operatorname{sech}\alpha\right)\sim\left(\frac{\sinh\left(2% \alpha\right)}{4\pi\nu}\right)^{\frac{1}{2}}e^{\nu(\tanh\alpha-\alpha)}\sum_{k% =0}^{\infty}\frac{V_{k}(\coth\alpha)}{\nu^{k}},$

$Y_{\nu}'\left(\nu\operatorname{sech}\alpha\right)\sim\left(\frac{\sinh\left(2% \alpha\right)}{\pi\nu}\right)^{\frac{1}{2}}e^{\nu(\alpha-\tanh\alpha)}\sum_{k=% 0}^{\infty}(-1)^{k}\frac{V_{k}(\coth\alpha)}{\nu^{k}}.$

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 $\xi=\nu(\tan\beta-\beta)-\tfrac{1}{4}\pi,$

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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

$J_{\nu}\left(\nu\sec\beta\right)\sim\left(\frac{2}{\pi\nu\tan\beta}\right)^{% \frac{1}{2}}\*\left(\cos\xi\sum_{k=0}^{\infty}\frac{U_{2k}(i\cot\beta)}{\nu^{2% k}}-i\sin\xi\sum_{k=0}^{\infty}\frac{U_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$

$Y_{\nu}\left(\nu\sec\beta\right)\sim\left(\frac{2}{\pi\nu\tan\beta}\right)^{% \frac{1}{2}}\*\left(\sin\xi\sum_{k=0}^{\infty}\frac{U_{2k}(i\cot\beta)}{\nu^{2% k}}+i\cos\xi\sum_{k=0}^{\infty}\frac{U_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$

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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, : 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

$J_{\nu}'\left(\nu\sec\beta\right)\sim\left(\frac{\sin\left(2\beta\right)}{\pi% \nu}\right)^{\frac{1}{2}}\*\left(-\sin\xi\sum_{k=0}^{\infty}\frac{V_{2k}(i\cot% \beta)}{\nu^{2k}}-i\cos\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\cot\beta)}{\nu^{% 2k+1}}\right),$

$Y_{\nu}'\left(\nu\sec\beta\right)\sim\left(\frac{\sin\left(2\beta\right)}{\pi% \nu}\right)^{\frac{1}{2}}\*\left(\cos\xi\sum_{k=0}^{\infty}\frac{V_{2k}(i\cot% \beta)}{\nu^{2k}}-i\sin\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\cot\beta)}{\nu^{% 2k+1}}\right).$

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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, : 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 of degree 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

$J_{\nu}\left(\nu+a\nu^{\frac{1}{3}}\right)\sim\frac{2^{\frac{1}{3}}}{\nu^{% \frac{1}{3}}}\operatorname{Ai}\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty% }\frac{P_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{2}{3}}}{\nu}\operatorname{Ai}'% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$ $|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta$ ,

$Y_{\nu}\left(\nu+a\nu^{\frac{1}{3}}\right)\sim-\frac{2^{\frac{1}{3}}}{\nu^{% \frac{1}{3}}}\operatorname{Bi}\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty% }\frac{P_{k}(a)}{\nu^{2k/3}}-\frac{2^{\frac{2}{3}}}{\nu}\operatorname{Bi}'% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$ $|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta$ .

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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, : 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 $\rselection{{H^{(1)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim\frac{2^{\frac{4}{3}}}{% \nu^{\frac{1}{3}}}e^{\mp\pi i/3}\operatorname{Ai}\left(e^{\mp\pi i/3}2^{\frac{% 1}{3}}a\right)\sum_{k=0}^{\infty}\frac{P_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{5}% {3}}}{\nu}e^{\pm\pi i/3}\operatorname{Ai}'\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a% \right)\sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$

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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, : 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

$P_{0}(a)=1,$

$P_{1}(a)=-\tfrac{1}{5}a,$

$P_{2}(a)=-\tfrac{9}{100}a^{5}+\tfrac{3}{35}a^{2},$

$P_{3}(a)=\tfrac{957}{7000}a^{6}-\tfrac{173}{3150}a^{3}-\tfrac{1}{225},$

$P_{4}(a)=\tfrac{27}{20000}a^{10}-\tfrac{23573}{1\;47000}a^{7}+\tfrac{5903}{1\;% 38600}a^{4}+\tfrac{947}{3\;46500}a,$

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Defines:: $P_{k}(a)$ : polynomial coefficient (locally)
Symbols:: : 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

$Q_{0}(a)=\tfrac{3}{10}a^{2},$

$Q_{1}(a)=-\tfrac{17}{70}a^{3}+\tfrac{1}{70},$

$Q_{2}(a)=-\tfrac{9}{1000}a^{7}+\tfrac{611}{3150}a^{4}-\tfrac{37}{3150}a,$

$Q_{3}(a)=\tfrac{549}{28000}a^{8}-\tfrac{1\;10767}{6\;93000}a^{5}+\tfrac{79}{12% 375}a^{2}.$

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Defines:: $Q_{k}(a)$ : polynomial coefficient (locally)
Symbols:: : 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

$J_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)\sim-\frac{2^{\frac{2}{3}}}{\nu^{% \frac{2}{3}}}\operatorname{Ai}'\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{% \infty}\frac{R_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}% \operatorname{Ai}\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(% a)}{\nu^{2k/3}},$ $|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta$ ,

$Y_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)\sim\frac{2^{\frac{2}{3}}}{\nu^{% \frac{2}{3}}}\operatorname{Bi}'\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{% \infty}\frac{R_{k}(a)}{\nu^{2k/3}}-\frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}% \operatorname{Bi}\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(% a)}{\nu^{2k/3}},$ $|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta$ .

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, : 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

$R_{0}(a)=1,$

$R_{1}(a)=-\tfrac{4}{5}a,$

$R_{2}(a)=-\tfrac{9}{100}a^{5}+\tfrac{57}{70}a^{2},$

$R_{3}(a)=\tfrac{699}{3500}a^{6}-\tfrac{2617}{3150}a^{3}+\tfrac{23}{3150},$

$R_{4}(a)=\tfrac{27}{20000}a^{10}-\tfrac{46631}{1\;47000}a^{7}+\tfrac{3889}{462% 0}a^{4}-\tfrac{1159}{1\;15500}a,$

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Defines:: $R_{k}(a)$ : polynomial coefficient (locally)
Symbols:: : 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

$S_{0}(a)=\tfrac{3}{5}a^{3}-\tfrac{1}{5},$

$S_{1}(a)=-\tfrac{131}{140}a^{4}+\tfrac{1}{5}a,$

$S_{2}(a)=-\tfrac{9}{500}a^{8}+\tfrac{5437}{4500}a^{5}-\tfrac{593}{3150}a^{2},$

$S_{3}(a)=\tfrac{369}{7000}a^{9}-\tfrac{9\;99443}{6\;93000}a^{6}+\tfrac{31727}{% 1\;73250}a^{3}+\tfrac{947}{3\;46500}.$

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Defines:: $S_{k}(a)$ : polynomial coefficient (locally)
Symbols:: : 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 (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