§10.19 Asymptotic Expansions for Large Order

§10.19(i) Asymptotic Forms

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

 10.19.1 $J_{\nu}\left(z\right)\sim\frac{1}{\sqrt{2\pi\nu}}\left(\frac{ez}{2\nu}\right)^% {\nu},$
 10.19.2 $Y_{\nu}\left(z\right)\sim-i{H^{(1)}_{\nu}}\left(z\right)\sim i{H^{(2)}_{\nu}}% \left(z\right)\sim-\sqrt{\frac{2}{\pi\nu}}\left(\frac{ez}{2\nu}\right)^{-\nu}.$

§10.19(ii) Debye’s Expansions

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

 10.19.3 $\displaystyle J_{\nu}\left(\nu\operatorname{sech}\alpha\right)$ $\displaystyle\sim\frac{e^{\nu(\tanh\alpha-\alpha)}}{(2\pi\nu\tanh\alpha)^{% \frac{1}{2}}}\sum_{k=0}^{\infty}\frac{U_{k}(\coth\alpha)}{\nu^{k}},$ $\displaystyle Y_{\nu}\left(\nu\operatorname{sech}\alpha\right)$ $\displaystyle\sim-\frac{e^{\nu(\alpha-\tanh\alpha)}}{(\tfrac{1}{2}\pi\nu\tanh% \alpha)^{\frac{1}{2}}}\*\sum_{k=0}^{\infty}(-1)^{k}\frac{U_{k}(\coth\alpha)}{% \nu^{k}},$
 10.19.4 $\displaystyle J_{\nu}'\left(\nu\operatorname{sech}\alpha\right)$ $\displaystyle\sim\left(\frac{\sinh\left(2\alpha\right)}{4\pi\nu}\right)^{\frac% {1}{2}}e^{\nu(\tanh\alpha-\alpha)}\sum_{k=0}^{\infty}\frac{V_{k}(\coth\alpha)}% {\nu^{k}},$ $\displaystyle Y_{\nu}'\left(\nu\operatorname{sech}\alpha\right)$ $\displaystyle\sim\left(\frac{\sinh\left(2\alpha\right)}{\pi\nu}\right)^{\frac{% 1}{2}}e^{\nu(\alpha-\tanh\alpha)}\sum_{k=0}^{\infty}(-1)^{k}\frac{V_{k}(\coth% \alpha)}{\nu^{k}}.$

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 $\displaystyle J_{\nu}\left(\nu\sec\beta\right)$ $\displaystyle\sim\left(\frac{2}{\pi\nu\tan\beta}\right)^{\frac{1}{2}}\*\left(% \cos\xi\sum_{k=0}^{\infty}\frac{U_{2k}(i\cot\beta)}{\nu^{2k}}-i\sin\xi\sum_{k=% 0}^{\infty}\frac{U_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$ $\displaystyle Y_{\nu}\left(\nu\sec\beta\right)$ $\displaystyle\sim\left(\frac{2}{\pi\nu\tan\beta}\right)^{\frac{1}{2}}\*\left(% \sin\xi\sum_{k=0}^{\infty}\frac{U_{2k}(i\cot\beta)}{\nu^{2k}}+i\cos\xi\sum_{k=% 0}^{\infty}\frac{U_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$
 10.19.7 $\displaystyle J_{\nu}'\left(\nu\sec\beta\right)$ $\displaystyle\sim\left(\frac{\sin\left(2\beta\right)}{\pi\nu}\right)^{\frac{1}% {2}}\*\left(-\sin\xi\sum_{k=0}^{\infty}\frac{V_{2k}(i\cot\beta)}{\nu^{2k}}-i% \cos\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right),$ $\displaystyle Y_{\nu}'\left(\nu\sec\beta\right)$ $\displaystyle\sim\left(\frac{\sin\left(2\beta\right)}{\pi\nu}\right)^{\frac{1}% {2}}\*\left(\cos\xi\sum_{k=0}^{\infty}\frac{V_{2k}(i\cot\beta)}{\nu^{2k}}-i% \sin\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\cot\beta)}{\nu^{2k+1}}\right).$

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

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 $\displaystyle J_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)$ $\displaystyle\sim-\frac{2^{\frac{2}{3}}}{\nu^{\frac{2}{3}}}\operatorname{Ai}'% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}+% \frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}\operatorname{Ai}\left(-2^{\frac{1}{3% }}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$ $|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta$, $\displaystyle Y_{\nu}'\left(\nu+a\nu^{\frac{1}{3}}\right)$ $\displaystyle\sim\frac{2^{\frac{2}{3}}}{\nu^{\frac{2}{3}}}\operatorname{Bi}'% \left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}-% \frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}\operatorname{Bi}\left(-2^{\frac{1}{3% }}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$ $|\operatorname{ph}\nu|\leq\tfrac{1}{2}\pi-\delta$.
 10.19.13 $\rselection{{H^{(1)}_{\nu}}'\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}'\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim-\frac{2^{\frac{5}{3}}% }{\nu^{\frac{2}{3}}}e^{\pm\pi i/3}\operatorname{Ai}'\left(e^{\mp\pi i/3}2^{% \frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}+\frac{2^{% \frac{4}{3}}}{\nu^{\frac{4}{3}}}e^{\mp\pi i/3}\operatorname{Ai}\left(e^{\mp\pi i% /3}2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$

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 $Y_{\nu}'\left(\nu\right)$.