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

## §10.19(ii) Debye’s Expansions

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

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

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 $\xi=\nu(\mathop{\tan\/}\nolimits\beta-\beta)-\tfrac{1}{4}\pi,$ Defines: $\xi$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\tan\/}\nolimits\NVar{z}$: 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)

then

 10.19.6 $\displaystyle\mathop{J_{\nu}\/}\nolimits\!\left(\nu\mathop{\sec\/}\nolimits% \beta\right)$ $\displaystyle\sim\left(\frac{2}{\pi\nu\mathop{\tan\/}\nolimits\beta}\right)^{% \frac{1}{2}}\*\left(\mathop{\cos\/}\nolimits\xi\sum_{k=0}^{\infty}\frac{U_{2k}% (i\mathop{\cot\/}\nolimits\beta)}{\nu^{2k}}-i\mathop{\sin\/}\nolimits\xi\sum_{% k=0}^{\infty}\frac{U_{2k+1}(i\mathop{\cot\/}\nolimits\beta)}{\nu^{2k+1}}\right),$ $\displaystyle\mathop{Y_{\nu}\/}\nolimits\!\left(\nu\mathop{\sec\/}\nolimits% \beta\right)$ $\displaystyle\sim\left(\frac{2}{\pi\nu\mathop{\tan\/}\nolimits\beta}\right)^{% \frac{1}{2}}\*\left(\mathop{\sin\/}\nolimits\xi\sum_{k=0}^{\infty}\frac{U_{2k}% (i\mathop{\cot\/}\nolimits\beta)}{\nu^{2k}}+i\mathop{\cos\/}\nolimits\xi\sum_{% k=0}^{\infty}\frac{U_{2k+1}(i\mathop{\cot\/}\nolimits\beta)}{\nu^{2k+1}}\right),$
 10.19.7 $\displaystyle\mathop{J_{\nu}\/}\nolimits'\!\left(\nu\mathop{\sec\/}\nolimits% \beta\right)$ $\displaystyle\sim\left(\frac{\mathop{\sin\/}\nolimits\!\left(2\beta\right)}{% \pi\nu}\right)^{\frac{1}{2}}\*\left(-\mathop{\sin\/}\nolimits\xi\sum_{k=0}^{% \infty}\frac{V_{2k}(i\mathop{\cot\/}\nolimits\beta)}{\nu^{2k}}-i\mathop{\cos\/% }\nolimits\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\mathop{\cot\/}\nolimits\beta)% }{\nu^{2k+1}}\right),$ $\displaystyle\mathop{Y_{\nu}\/}\nolimits'\!\left(\nu\mathop{\sec\/}\nolimits% \beta\right)$ $\displaystyle\sim\left(\frac{\mathop{\sin\/}\nolimits\!\left(2\beta\right)}{% \pi\nu}\right)^{\frac{1}{2}}\*\left(\mathop{\cos\/}\nolimits\xi\sum_{k=0}^{% \infty}\frac{V_{2k}(i\mathop{\cot\/}\nolimits\beta)}{\nu^{2k}}-i\mathop{\sin\/% }\nolimits\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\mathop{\cot\/}\nolimits\beta)% }{\nu^{2k+1}}\right).$

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 $\displaystyle\mathop{J_{\nu}\/}\nolimits\!\left(\nu+a\nu^{\frac{1}{3}}\right)$ $\displaystyle\sim\frac{2^{\frac{1}{3}}}{\nu^{\frac{1}{3}}}\mathop{\mathrm{Ai}% \/}\nolimits\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{\mathop{P% _{k}\/}\nolimits\!\left(a\right)}{\nu^{2k/3}}+\frac{2^{\frac{2}{3}}}{\nu}% \mathop{\mathrm{Ai}\/}\nolimits'\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{% \infty}\frac{\mathop{Q_{k}\/}\nolimits\!\left(a\right)}{\nu^{2k/3}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$, $\displaystyle\mathop{Y_{\nu}\/}\nolimits\!\left(\nu+a\nu^{\frac{1}{3}}\right)$ $\displaystyle\sim-\frac{2^{\frac{1}{3}}}{\nu^{\frac{1}{3}}}\mathop{\mathrm{Bi}% \/}\nolimits\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{\mathop{P% _{k}\/}\nolimits\!\left(a\right)}{\nu^{2k/3}}-\frac{2^{\frac{2}{3}}}{\nu}% \mathop{\mathrm{Bi}\/}\nolimits'\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{% \infty}\frac{\mathop{Q_{k}\/}\nolimits\!\left(a\right)}{\nu^{2k/3}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$.

Also,

 10.19.9 $\rselection{\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(\nu+a\nu^{\frac{1}{3}}% \right)\\ \mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim% \frac{2^{\frac{4}{3}}}{\nu^{\frac{1}{3}}}e^{\mp\pi i/3}\mathop{\mathrm{Ai}\/}% \nolimits\!\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac% {\mathop{P_{k}\/}\nolimits\!\left(a\right)}{\nu^{2k/3}}+\frac{2^{\frac{5}{3}}}% {\nu}e^{\pm\pi i/3}\mathop{\mathrm{Ai}\/}\nolimits'\!\left(e^{\mp\pi i/3}2^{% \frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{\mathop{Q_{k}\/}\nolimits\!\left(% a\right)}{\nu^{2k/3}},$

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

 10.19.10 $\displaystyle\mathop{P_{0}\/}\nolimits\!\left(a\right)$ $\displaystyle=1,$ $\displaystyle\mathop{P_{1}\/}\nolimits\!\left(a\right)$ $\displaystyle=-\tfrac{1}{5}a,$ $\displaystyle\mathop{P_{2}\/}\nolimits\!\left(a\right)$ $\displaystyle=-\tfrac{9}{100}a^{5}+\tfrac{3}{35}a^{2},$ $\displaystyle\mathop{P_{3}\/}\nolimits\!\left(a\right)$ $\displaystyle=\tfrac{957}{7000}a^{6}-\tfrac{173}{3150}a^{3}-\tfrac{1}{225},$ $\displaystyle\mathop{P_{4}\/}\nolimits\!\left(a\right)$ $\displaystyle=\tfrac{27}{20000}a^{10}-\tfrac{23573}{1\;47000}a^{7}+\tfrac{5903% }{1\;38600}a^{4}+\tfrac{947}{3\;46500}a,$ Symbols: $\mathop{P_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)=\mathop{P^{0}_{\nu}% \/}\nolimits\!\left(z\right)$: Legendre function of the first kind 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 See also: Annotations for 10.19(iii)
 10.19.11 $\displaystyle\mathop{Q_{0}\/}\nolimits\!\left(a\right)$ $\displaystyle=\tfrac{3}{10}a^{2},$ $\displaystyle\mathop{Q_{1}\/}\nolimits\!\left(a\right)$ $\displaystyle=-\tfrac{17}{70}a^{3}+\tfrac{1}{70},$ $\displaystyle\mathop{Q_{2}\/}\nolimits\!\left(a\right)$ $\displaystyle=-\tfrac{9}{1000}a^{7}+\tfrac{611}{3150}a^{4}-\tfrac{37}{3150}a,$ $\displaystyle\mathop{Q_{3}\/}\nolimits\!\left(a\right)$ $\displaystyle=\tfrac{549}{28000}a^{8}-\tfrac{1\;10767}{6\;93000}a^{5}+\tfrac{7% 9}{12375}a^{2}.$ Symbols: $\mathop{Q_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)=\mathop{Q^{0}_{\nu}% \/}\nolimits\!\left(z\right)$: Legendre function of the second kind A&S Ref: 9.3.26 Referenced by: 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 Errata (effective with 1.0.10): Originally the first term on the right-hand side of the equation for $\mathop{Q_{3}\/}\nolimits\!\left(a\right)$ was written incorrectly as $-\tfrac{549}{28000}a^{8}$. Reported 2015-03-16 by Svante Janson See also: Annotations for 10.19(iii)
 10.19.12 $\displaystyle\mathop{J_{\nu}\/}\nolimits'\!\left(\nu+a\nu^{\frac{1}{3}}\right)$ $\displaystyle\sim-\frac{2^{\frac{2}{3}}}{\nu^{\frac{2}{3}}}\mathop{\mathrm{Ai}% \/}\nolimits'\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)% }{\nu^{2k/3}}+\frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}\mathop{\mathrm{Ai}\/}% \nolimits\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{% \nu^{2k/3}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$, $\displaystyle\mathop{Y_{\nu}\/}\nolimits'\!\left(\nu+a\nu^{\frac{1}{3}}\right)$ $\displaystyle\sim\frac{2^{\frac{2}{3}}}{\nu^{\frac{2}{3}}}\mathop{\mathrm{Bi}% \/}\nolimits'\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)% }{\nu^{2k/3}}-\frac{2^{\frac{1}{3}}}{\nu^{\frac{4}{3}}}\mathop{\mathrm{Bi}\/}% \nolimits\!\left(-2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{% \nu^{2k/3}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$.
 10.19.13 $\rselection{\mathop{{H^{(1)}_{\nu}}\/}\nolimits'\!\left(\nu+a\nu^{\frac{1}{3}}% \right)\\ \mathop{{H^{(2)}_{\nu}}\/}\nolimits'\!\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim% -\frac{2^{\frac{5}{3}}}{\nu^{\frac{2}{3}}}e^{\pm\pi i/3}\mathop{\mathrm{Ai}\/}% \nolimits'\!\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}% \frac{R_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{4}{3}}}{\nu^{\frac{4}{3}}}e^{\mp\pi i% /3}\mathop{\mathrm{Ai}\/}\nolimits\!\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a\right% )\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$

with sectors of validity $-\tfrac{1}{2}\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits\nu\leq\tfrac{3}{2}% \pi-\delta$ and $-\tfrac{3}{2}\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits\nu\leq\tfrac{1}{2}% \pi-\delta$, 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 (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)
 10.19.15 $\displaystyle S_{0}(a)$ $\displaystyle=\tfrac{3}{5}a^{3}-\tfrac{1}{5},$ $\displaystyle S_{1}(a)$ $\displaystyle=-\tfrac{131}{140}a^{4}+\tfrac{1}{5}a,$ $\displaystyle S_{2}(a)$ $\displaystyle=-\tfrac{9}{500}a^{8}+\tfrac{5437}{4500}a^{5}-\tfrac{593}{3150}a^% {2},$ $\displaystyle S_{3}(a)$ $\displaystyle=\tfrac{369}{7000}a^{9}-\tfrac{9\;99443}{6\;93000}a^{6}+\tfrac{31% 727}{1\;73250}a^{3}+\tfrac{947}{3\;46500}.$ Defines: $S_{k}(a)$: polynomial (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)

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 $\mathop{J_{\nu}\/}\nolimits\!\left(\nu\right)$ and $\mathop{Y_{\nu}\/}\nolimits\!\left(\nu\right)$), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). The last reference also includes the corresponding expansions for $\mathop{J_{\nu}\/}\nolimits'\!\left(\nu\right)$ and $\mathop{Y_{\nu}\/}\nolimits'\!\left(\nu\right)$.