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

Keywords:: Bessel functions, Hankel functions
Permalink:: http://dlmf.nist.gov/10.19

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

§10.19(i) Asymptotic Forms

Notes:: These results follow from (10.2.2), (10.2.3), (10.4.3), (10.8.1), (5.5.3), (5.11.3).
Keywords:: Bessel functions, Hankel functions, derivatives
Referenced by:: ¶ ‣ §3.6(vi)
Permalink:: http://dlmf.nist.gov/10.19.i

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\sim$ : asymptotic equality, : 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

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

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), : base of exponential function, $\sim$ : asymptotic equality, : 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

§10.19(ii) Debye’s Expansions

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).
Keywords:: Bessel functions, derivatives
Referenced by:: §11.11(iii)
Permalink:: http://dlmf.nist.gov/10.19.ii

If $\nu\to\infty$ through positive real values with $\alpha$ fixed, then

10.19.3

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

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

10.19.4

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

${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu\mathop{\mathrm{sech}\/}% \nolimits\alpha\right)\sim\left(\frac{\mathop{\sinh\/}\nolimits\!\left(2\alpha% \right)}{\pi\nu}\right)^{{\frac{1}{2}}}e^{{\nu(\alpha-\mathop{\tanh\/}% \nolimits\alpha)}}\sum_{{k=0}}^{\infty}(-1)^{k}\frac{V_{k}(\mathop{\coth\/}% \nolimits\alpha)}{\nu^{k}}.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, : base of exponential function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\sim$ : asymptotic equality, : nonnegative integer, $\nu$ : complex parameter and $V_{k}(p)$ : polynomials
A&S Ref:: 9.3.11, 9.3.12
Permalink:: http://dlmf.nist.gov/10.19.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

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

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function, $\nu$ : complex parameter, $\beta$ and $\xi$
Permalink:: http://dlmf.nist.gov/10.19.E5
Encodings:: TeX, pMML, png

then

10.19.6

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

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

10.19.7

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

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

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

For error bounds for the first of (10.19.3) see Olver (1997b, p. 382).

§10.19(iii) Transition Region

Keywords:: Bessel functions, Hankel functions, derivatives
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/10.19.iii
Addition (effective with 1.0.5):: The reference to Jentschura and Lötstedt (2012) has been added at the end of this subsection.
Reported 2012-07-30

As $\nu\to\infty$ , with $a(\in\Complex)$ fixed,

10.19.8

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

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

Also,

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : base of exponential function, $\sim$ : asymptotic equality, : nonnegative integer and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.19.E9
Encodings:: TeX, pMML, png

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

$\mathop{P_{{0}}\/}\nolimits\!\left(a\right)=1,$

$\mathop{P_{{1}}\/}\nolimits\!\left(a\right)=-\tfrac{1}{5}a,$

$\mathop{P_{{2}}\/}\nolimits\!\left(a\right)=-\tfrac{9}{100}a^{5}+\tfrac{3}{35}% a^{2},$

$\mathop{P_{{3}}\/}\nolimits\!\left(a\right)=\tfrac{957}{7000}a^{6}-\tfrac{173}% {3150}a^{3}-\tfrac{1}{225},$

$\mathop{P_{{4}}\/}\nolimits\!\left(a\right)=\tfrac{27}{20000}a^{{10}}-\tfrac{2% 3573}{1\;47000}a^{7}+\tfrac{5903}{1\;38600}a^{4}+\tfrac{947}{3\;46500}a,$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated 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

10.19.11

$\mathop{Q_{{0}}\/}\nolimits\!\left(a\right)=\tfrac{3}{10}a^{2},$

$\mathop{Q_{{1}}\/}\nolimits\!\left(a\right)=-\tfrac{17}{70}a^{3}+\tfrac{1}{70},$

$\mathop{Q_{{2}}\/}\nolimits\!\left(a\right)=-\tfrac{9}{1000}a^{7}+\tfrac{611}{% 3150}a^{4}-\tfrac{37}{3150}a,$

$\mathop{Q_{{3}}\/}\nolimits\!\left(a\right)=-\tfrac{549}{28000}a^{8}-\tfrac{1% \;10767}{6\;93000}a^{5}+\tfrac{79}{12375}a^{2}.$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind
A&S Ref:: 9.3.26
Permalink:: http://dlmf.nist.gov/10.19.E11
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

10.19.12

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

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, : nonnegative integer, $\nu$ : complex parameter, $\delta$ : small positive constant, $R_{k}(a)$ : polynomial and $S_{k}(a)$ : polynomial
A&S Ref:: 9.3.27, 9.3.28
Permalink:: http://dlmf.nist.gov/10.19.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

10.19.13 $\rselection{{\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(\nu+a% \nu^{{\frac{1}{3}}}\right)\\ {\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(\nu+a\nu^{{\frac{1% }{3}}}\right)}\sim-\frac{2^{{\frac{5}{3}}}}{\nu^{{\frac{2}{3}}}}e^{{\pm\pi i/3% }}{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(e^{{\mp\pi i/3}}2^{{% \frac{1}{3}}}a\right)\sum_{{k=0}}^{\infty}\frac{R_{k}(a)}{\nu^{{2k/3}}}+\frac{% 2^{{\frac{4}{3}}}}{\nu^{{\frac{4}{3}}}}e^{{\mp\pi i/3}}\mathop{\mathrm{Ai}\/}% \nolimits\!\left(e^{{\mp\pi i/3}}2^{{\frac{1}{3}}}a\right)\sum_{{k=0}}^{\infty% }\frac{S_{k}(a)}{\nu^{{2k/3}}},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), : base of exponential function, $\sim$ : asymptotic equality, : nonnegative integer, $\nu$ : complex parameter, $R_{k}(a)$ : polynomial and $S_{k}(a)$ : polynomial
Permalink:: http://dlmf.nist.gov/10.19.E13
Encodings:: TeX, pMML, png

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

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

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

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

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

$R_{4}(a)=\tfrac{27}{20000}a^{{10}}-\tfrac{46631}{1\;47000}a^{7}+\tfrac{3889}{4% 620}a^{4}-\tfrac{1159}{1\;15500}a,$

Symbols:: : nonnegative integer and $R_{k}(a)$ : polynomial
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

10.19.15

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

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

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

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

Symbols:: : nonnegative integer and $S_{k}(a)$ : polynomial
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

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 $\mathop{J_{{\nu}}\/}\nolimits\!\left(\nu\right)$ 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 ${\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu\right)$ and ${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\nu\right)$ .

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.18 Modulus and Phase Functions 10.20 Uniform Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

Contents

§10.19(i) Asymptotic Forms

§10.19(ii) Debye’s Expansions

§10.19(iii) Transition Region