11 Struve and Related Functions Related Functions 11.10 Anger–Weber Functions 11.12 Physical Applications

§11.11 Asymptotic Expansions of Anger–Weber Functions

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§11.11(i) Large , Fixed $\nu$
§11.11(ii) Large $|\nu|$ , Fixed
§11.11(iii) Large $\nu$ , Fixed $z/\nu$

§11.11(i) Large , Fixed $\nu$

Notes:: See Watson (1944, §10.14). (11.11.2), (11.11.3) follow from (11.10.15), (11.10.16).
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.11.i

Let $F_{0}(\nu)=G_{0}(\nu)=1$ , and for $k=1,2,3,\dots$ ,

11.11.1

$\displaystyle F_{{\mathrm{k}}}(\nu)=(\nu^{2}-1^{2})(\nu^{2}-3^{2})\cdots(\nu^{% 2}-(2k-1)^{2}),$

$\displaystyle G_{{\mathrm{k}}}(\nu)=(\nu^{2}-2^{2})(\nu^{2}-4^{2})\cdots(\nu^{% 2}-(2k)^{2}).$

Symbols:: $\nu$ : real or complex order, : nonnegative integer, $F_{{\mathrm{k}}}(\nu)$ : expansion functions and $G_{{\mathrm{k}}}(\nu)$ : expansion functions
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Then as $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ( $<\pi$ )

11.11.2 $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)\sim\mathop{J_{{\nu}}\/}% \nolimits\!\left(z\right)\\ +\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}{\pi z}\left(\sum_{{k=0}}% ^{\infty}\frac{F_{k}(\nu)}{z^{{2k}}}-\frac{\nu}{z}\sum_{{k=0}}^{\infty}\frac{G% _{k}(\nu)}{z^{{2k}}}\right),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, $\nu$ : real or complex order, : nonnegative integer, $F_{{\mathrm{k}}}(\nu)$ : expansion functions and $G_{{\mathrm{k}}}(\nu)$ : expansion functions
Referenced by:: §11.11(i)
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11.11.3 $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)\sim-\mathop{Y_{{\nu}}\/% }\nolimits\!\left(z\right)-\frac{1+\mathop{\cos\/}\nolimits\!\left(\pi\nu% \right)}{\pi z}\sum_{{k=0}}^{\infty}\frac{F_{k}(\nu)}{z^{{2k}}}-\frac{\nu(1-% \mathop{\cos\/}\nolimits\!\left(\pi\nu\right))}{\pi z^{2}}\sum_{{k=0}}^{\infty% }\frac{G_{k}(\nu)}{z^{{2k}}},$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\sim$ : asymptotic equality, : complex variable, $\nu$ : real or complex order, : nonnegative integer, $F_{{\mathrm{k}}}(\nu)$ : expansion functions and $G_{{\mathrm{k}}}(\nu)$ : expansion functions
Referenced by:: §11.11(i)
Permalink:: http://dlmf.nist.gov/11.11.E3
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11.11.4 $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi z}\sum_% {{k=0}}^{\infty}\frac{F_{k}(\nu)}{z^{{2k}}}-\frac{\nu}{\pi z^{2}}\sum_{{k=0}}^% {\infty}\frac{G_{k}(\nu)}{z^{{2k}}}.$

Symbols:: $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, $\sim$ : asymptotic equality, : complex variable, $\nu$ : real or complex order, : nonnegative integer, $F_{{\mathrm{k}}}(\nu)$ : expansion functions and $G_{{\mathrm{k}}}(\nu)$ : expansion functions
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§11.11(ii) Large $|\nu|$ , Fixed

Notes:: The approximations for large $|\nu|$ and fixed follow from (11.10.8)–(11.10.11). Eqs. (11.11.7) follow from (11.6.5).
Keywords:: Anger–Weber functions
Referenced by:: ¶ ‣ §3.6(vi)
Permalink:: http://dlmf.nist.gov/11.11.ii

If is fixed, and $\nu\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\pi$ in such a way that $\nu$ is bounded away from the set of all integers, then

11.11.5 $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\sin\/}% \nolimits\!\left(\pi\nu\right)}{\pi\nu}\left(1-\frac{\nu z}{\nu^{2}-1}+\mathop% {O\/}\nolimits\!\left(\frac{1}{\nu^{2}}\right)\right),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and $\nu$ : real or complex order
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11.11.6 $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi\nu}\left({% \mathop{\sin\/}\nolimits^{{2}}}\!\left(\tfrac{1}{2}\pi\nu\right)+\frac{\nu z}{% \nu^{2}-1}{\mathop{\cos\/}\nolimits^{{2}}}\!\left(\tfrac{1}{2}\pi\nu\right)+% \mathop{O\/}\nolimits\!\left(\frac{1}{\nu^{2}}\right)\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and $\nu$ : real or complex order
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If $\nu=n(\in\Integer)$ , then (11.10.29) applies for $\mathop{\mathbf{J}_{{n}}\/}\nolimits\!\left(z\right)$ , and

11.11.7

$\displaystyle\mathop{\mathbf{E}_{{2n}}\/}\nolimits\!\left(z\right)\sim\frac{2z% }{(4n^{2}-1)\pi},$

$\displaystyle\mathop{\mathbf{E}_{{2n+1}}\/}\nolimits\!\left(z\right)\sim\frac{% 2}{(2n+1)\pi},$

$n\to\pm\infty$ .

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\sim$ : asymptotic equality, : complex variable and : integer order
Referenced by:: §11.11(ii)
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§11.11(iii) Large $\nu$ , Fixed $z/\nu$

Notes:: For (11.11.8)–(11.11.10), see Watson (1944, §10.15). For (11.11.11), see Dingle (1973, p. 388). For (11.11.8)–(11.11.19), see Olver (1997b, pp. 103 and 352).
Permalink:: http://dlmf.nist.gov/11.11.iii

For fixed $\lambda$ ,

11.11.8 $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim\frac{1}{% \pi}\sum_{{k=0}}^{\infty}\frac{(2k)!\,a_{k}(\lambda)}{\nu^{{2k+1}}},$ $\nu\to\infty$ , $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\pi-\delta$ ( $<\pi$ ),

Symbols:: $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, : : factorial, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\nu$ : real or complex order, : nonnegative integer, $\delta$ : arbitrary small positive constant, $\lambda$ : parameter and $a_{k}(\lambda)$ : expansion function
Referenced by:: §11.11(iii)
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where

11.11.9

$a_{0}=\frac{1}{1+\lambda},$

$a_{1}=-\frac{\lambda}{2(1+\lambda)^{4}},$

$a_{2}=\frac{9\lambda^{2}-\lambda}{24(1+\lambda)^{7}},$

$a_{3}=-\frac{225\lambda^{3}-54\lambda^{2}+\lambda}{720(1+\lambda)^{{10}}}.$

Symbols:: $\lambda$ : parameter and $a_{k}(\lambda)$ : expansion function
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For fixed $\lambda(>1)$ ,

11.11.10 $\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim-\frac{1}{% \pi}\sum_{{k=0}}^{\infty}\frac{(2k)!\,a_{k}(-\lambda)}{\nu^{{2k+1}}},$ $\nu\to+\infty$ .

Symbols:: $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, : : factorial, $\sim$ : asymptotic equality, $\nu$ : real or complex order, : nonnegative integer, $\lambda$ : parameter and $a_{k}(\lambda)$ : expansion function
Referenced by:: §11.11(iii), §11.11(iii)
Permalink:: http://dlmf.nist.gov/11.11.E10
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For fixed $\lambda$ , $0<\lambda<1$ ,

11.11.11 $\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim\sqrt{% \frac{2}{\pi\nu}}\,e^{{-\nu\mu}}\sum_{{k=0}}^{\infty}\frac{(\tfrac{1}{2})_{k}b% _{k}(\lambda)}{\nu^{{k}}},$ $\nu\to+\infty$ ,

Symbols:: $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, : base of exponential function, $\sim$ : asymptotic equality, $\nu$ : real or complex order, : nonnegative integer, $\lambda$ : parameter and $b_{k}(\lambda)$ : expansion function
Referenced by:: §11.11(iii)
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where

11.11.12 $\mu=\sqrt{1-\lambda^{2}}-\mathop{\ln\/}\nolimits\!\left(\frac{1+\sqrt{1-% \lambda^{2}}}{\lambda}\right),$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $\lambda$ : parameter
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and

11.11.13

$b_{0}(\lambda)=\frac{1}{(1-\lambda^{2})^{{1/4}}}$ ,

$b_{1}(\lambda)=\frac{2+3\lambda^{2}}{12(1-\lambda^{2})^{{7/4}}},$

$b_{2}(\lambda)=\frac{4+300\lambda^{2}+81\lambda^{4}}{864(1-\lambda^{2})^{{13/4% }}}$ .

Symbols:: $\lambda$ : parameter and $b_{k}(\lambda)$ : expansion function
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In particular, as $\nu\to+\infty$ ,

11.11.14 $\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim\frac{1}{% \pi\nu(\lambda-1)},$ $\lambda>1$ ,

Symbols:: $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, $\sim$ : asymptotic equality, $\nu$ : real or complex order and $\lambda$ : parameter
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11.11.15 $\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(\lambda\nu\right)\sim\left(% \frac{2}{\pi\nu}\right)^{{1/2}}\left(\frac{1+\sqrt{1-\lambda^{2}}}{\lambda}% \right)^{\nu}\frac{e^{{-\nu\sqrt{1-\lambda^{2}}}}}{(1-\lambda^{2})^{{1/4}}},$ $0<\lambda<1$ .

Symbols:: $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, : base of exponential function, $\sim$ : asymptotic equality, $\nu$ : real or complex order and $\lambda$ : parameter
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Also, as $\nu\to+\infty$ ,

11.11.16 $\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(\nu\right)\sim\frac{2^{{4/3}}}{% 3^{{7/6}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)\nu^{{1/3}}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, $\sim$ : asymptotic equality and $\nu$ : real or complex order
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and

11.11.17 $\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(\nu+a\nu^{{1/3}}\right)=2^{{1/3% }}\nu^{{-1/3}}\mathop{\mathrm{Hi}\/}\nolimits\!\left(-2^{{1/3}}a\right)+% \mathop{O\/}\nolimits\!\left(\nu^{{-1}}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function), $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, $\nu$ : real or complex order and $a_{k}(\lambda)$ : expansion function
Referenced by:: §11.11(iii)
Permalink:: http://dlmf.nist.gov/11.11.E17
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uniformly for bounded real values of . For the Scorer function $\mathop{\mathrm{Hi}\/}\nolimits$ see §9.12(i).

All of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–357) for $\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(\lambda\nu\right)$ as $\nu\to+\infty$ , one being uniform for $\delta\leq\lambda\leq 1$ , where $\delta$ again denotes an arbitrary small positive constant, and the other being uniform for $1\leq\lambda<\infty$ . (Note that Olver’s definition of $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ omits the factor $1/\pi$ in (11.10.4).) See also Watson (1944, §10.15).

Lastly, corresponding asymptotic approximations and expansions for $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(\lambda\nu\right)$ and $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(\lambda\nu\right)$ follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ ; see §10.19(ii). In particular,

11.11.18 $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(\nu\right)\sim\frac{2^{{1/3}}}{3% ^{{2/3}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)\nu^{{1/3}}},$ $\nu\to+\infty$ ,

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\sim$ : asymptotic equality and $\nu$ : real or complex order
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11.11.19 $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(\nu\right)\sim\frac{2^{{1/3}}}{3% ^{{7/6}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)\nu^{{1/3}}},$ $\nu\to+\infty$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\sim$ : asymptotic equality and $\nu$ : real or complex order
Referenced by:: §11.11(iii)
Permalink:: http://dlmf.nist.gov/11.11.E19
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© 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. 11.10 Anger–Weber Functions 11.12 Physical Applications

§11.11 Asymptotic Expansions of Anger–Weber Functions

Contents

§11.11(i) Large , Fixed

§11.11(ii) Large , Fixed

§11.11(iii) Large , Fixed

§11.11(i) Large , Fixed $\nu$

§11.11(ii) Large $|\nu|$ , Fixed

§11.11(iii) Large $\nu$ , Fixed $z/\nu$