§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 $z$ $(\neq 0)$ fixed, then

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

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

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), $e$ : base of exponential function, $\sim$ : asymptotic equality, $z$ : 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

§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 $\alpha$ $(>0)$ 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, $e$ : 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, $k$ : 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 $p$ of degree $3k$ 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
Permalink:: http://dlmf.nist.gov/10.19.iii

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, $e$ : base of exponential function, $\sim$ : asymptotic equality, $k$ : 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

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

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

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

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

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

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, $k$ : 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), $e$ : base of exponential function, $\sim$ : asymptotic equality, $k$ : 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

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

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

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

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

$R_{4}(a)=\tfrac{27}{20000}a^{{10}}-\tfrac{46631}{1\; 47000}a^{7}+\tfrac{3889}{4620}a^{4}-\tfrac{1159}{1\; 15500}a,$

Symbols:: $k$ : 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:: $k$ : 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 $a=0$ (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) and Temme (1997).

§10.19 Asymptotic Expansions for Large Order

Contents

§10.19(i) Asymptotic Forms

§10.19(ii) Debye’s Expansions

§10.19(iii) Transition Region