§11.11 Asymptotic Expansions of Anger–Weber Functions

Let $F_0(\nu )=G_0(\nu )=1$, and for $k=1,2,3,\mathrm{\dots }$,

11.11.1		$F_k(\nu )$	$=({\nu }^2-1^2)({\nu }^2-3^2)\mathrm{\cdots }({\nu }^2-{(2k-1)}^2),$
		$G_k(\nu )$	$=({\nu }^2-2^2)({\nu }^2-4^2)\mathrm{\cdots }({\nu }^2-{(2k)}^2).$
ⓘ Symbols: $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions Referenced by: Erratum (V1.0.11) for Clarifications Permalink: http://dlmf.nist.gov/11.11.E1 Encodings: TeX, TeX, pMML, pMML, png, png Correction) (effective with 1.0.11): Originally the subscript $k$ on the left-hand side was written in an incorrect font as $\mathrm{k}$. See also: Annotations for §11.11(i), §11.11 and Ch.11

Then as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi -\delta$ ()

11.11.2		$${\mathbf{J}}_{\nu }\left(z\right)\sim J_{\nu }\left(z\right)+\frac{\sin \left(\pi \nu \right)}{\pi z}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu }{z}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}}\right),$$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions Referenced by: §11.11(i) Permalink: http://dlmf.nist.gov/11.11.E2 Encodings: TeX, pMML, png See also: Annotations for §11.11(i), §11.11 and Ch.11

11.11.3		$${\mathbf{E}}_{\nu }\left(z\right)\sim -Y_{\nu }\left(z\right)-\frac{1+\cos \left(\pi \nu \right)}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu (1-\cos \left(\pi \nu \right))}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}},$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions Referenced by: §11.11(i) Permalink: http://dlmf.nist.gov/11.11.E3 Encodings: TeX, pMML, png See also: Annotations for §11.11(i), §11.11 and Ch.11

11.11.4		$${\mathbf{A}}_{\nu }\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu }{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}}.$$
ⓘ Symbols: ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions Permalink: http://dlmf.nist.gov/11.11.E4 Encodings: TeX, pMML, png See also: Annotations for §11.11(i), §11.11 and Ch.11

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

11.11.5		$${\mathbf{J}}_{\nu }\left(z\right)=\frac{\sin \left(\pi \nu \right)}{\pi \nu }\left(1-\frac{\nu z}{{\nu }^2-1}+O\left(\frac{1}{{\nu }^2}\right)\right),$$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.11.E5 Encodings: TeX, pMML, png See also: Annotations for §11.11(ii), §11.11 and Ch.11

11.11.6		$${\mathbf{E}}_{\nu }\left(z\right)=\frac{2}{\pi \nu }\left({\sin }^2\left(\frac{1}{2}\pi \nu \right)+\frac{\nu z}{{\nu }^2-1}{\cos }^2\left(\frac{1}{2}\pi \nu \right)+O\left(\frac{1}{{\nu }^2}\right)\right).$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.11.E6 Encodings: TeX, pMML, png See also: Annotations for §11.11(ii), §11.11 and Ch.11

If $\nu =n(\in \mathbb{Z})$, then (11.10.29) applies for ${\mathbf{J}}_n\left(z\right)$, and

11.11.7		${\mathbf{E}}_{2n}\left(z\right)$	$\sim {\displaystyle \frac{2z}{(4n^2-1)\pi }},$
		${\mathbf{E}}_{2n+1}\left(z\right)$	$\sim {\displaystyle \frac{2}{(2n+1)\pi }},$
$n\to \pm \mathrm{\infty }$.
ⓘ Symbols: ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $n$: integer order Referenced by: §11.11(ii) Permalink: http://dlmf.nist.gov/11.11.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.11(ii), §11.11 and Ch.11

For fixed $\lambda$ $(>0)$,

11.11.8		$${\mathbf{A}}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$
$\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$ (),
ⓘ Symbols: ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function Referenced by: §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E8 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

where

11.11.9		$a_0(\lambda )$	$={\displaystyle \frac{1}{1+\lambda }},$
		$a_1(\lambda )$	$=-{\displaystyle \frac{\lambda }{2{(1+\lambda )}^4}},$
		$a_2(\lambda )$	$={\displaystyle \frac{9{\lambda }^2-\lambda }{24{(1+\lambda )}^7}},$
		$a_3(\lambda )$	$=-{\displaystyle \frac{225{\lambda }^3-54{\lambda }^2+\lambda }{720{(1+\lambda )}^{10}}}.$
ⓘ Symbols: $\lambda$: parameter and $a_k(\lambda )$: expansion function Referenced by: Erratum (V1.0.11) for Clarifications Permalink: http://dlmf.nist.gov/11.11.E9 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png Clarification (effective with 1.0.11): Originally the argument $\lambda$ of these functions was suppressed; as a notational clarification, the argument has been made explicit. See also: Annotations for §11.11(iii), §11.11 and Ch.11

For fixed $\lambda (>1)$,

11.11.10		$${\mathbf{A}}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$
$\nu \to +\mathrm{\infty }$.
ⓘ Symbols: ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\nu$: real or complex order, $k$: 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 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

For fixed $\lambda$, ,

11.11.11		$${\mathbf{A}}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\mathrm{e}}^{-\nu \mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2})}_k{b}_k(\lambda )}{{\nu }^k},$$
$\nu \to +\mathrm{\infty }$,
ⓘ Symbols: ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\nu$: real or complex order, $k$: nonnegative integer, $\lambda$: parameter and $b_k(\lambda )$: expansion function Referenced by: §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E11 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

where

11.11.12		$$\mu =\sqrt{1-{\lambda }^2}-\ln \left(\frac{1+\sqrt{1-{\lambda }^2}}{\lambda }\right),$$
ⓘ Symbols: $\ln z$: principal branch of logarithm function and $\lambda$: parameter Permalink: http://dlmf.nist.gov/11.11.E12 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

and

11.11.13		$b_0(\lambda )$	$={\displaystyle \frac{1}{{(1-{\lambda }^2)}^{1/4}}}$,
		$b_1(\lambda )$	$={\displaystyle \frac{2+3{\lambda }^2}{12{(1-{\lambda }^2)}^{7/4}}},$
		$b_2(\lambda )$	$={\displaystyle \frac{4+300{\lambda }^2+81{\lambda }^4}{864{(1-{\lambda }^2)}^{13/4}}}$.
ⓘ Symbols: $\lambda$: parameter and $b_k(\lambda )$: expansion function Permalink: http://dlmf.nist.gov/11.11.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

In particular, as $\nu \to +\mathrm{\infty }$,

11.11.14		$${\mathbf{A}}_{-\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi \nu (\lambda -1)},$$
$\lambda >1$,
ⓘ Symbols: ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\nu$: real or complex order and $\lambda$: parameter Permalink: http://dlmf.nist.gov/11.11.E14 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

11.11.15		$${\mathbf{A}}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\left(\frac{1+\sqrt{1-{\lambda }^2}}{\lambda }\right)}^{\nu }\frac{{\mathrm{e}}^{-\nu \sqrt{1-{\lambda }^2}}}{{(1-{\lambda }^2)}^{1/4}},$$
.
ⓘ Symbols: ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\nu$: real or complex order and $\lambda$: parameter Permalink: http://dlmf.nist.gov/11.11.E15 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

Also, as $\nu \to +\mathrm{\infty }$,

11.11.16		$${\mathbf{A}}_{-\nu }\left(\nu \right)\sim \frac{2^{4/3}}{3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: asymptotic equality and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.11.E16 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

and

11.11.17		$${\mathbf{A}}_{-\nu }\left(\nu +a{\nu }^{1/3}\right)=2^{1/3}{\nu }^{-1/3}\mathrm{Hi}\left(-2^{1/3}a\right)+O\left({\nu }^{-1}\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), ${\mathbf{A}}_{\nu }\left(z\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

uniformly for bounded real values of $a$. For the Scorer function $\mathrm{Hi}$ 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 ${\mathbf{A}}_{-\nu }\left(\lambda \nu \right)$ as $\nu \to +\mathrm{\infty }$, one being uniform for $\delta \le \lambda \le 1$, where $\delta$ again denotes an arbitrary small positive constant, and the other being uniform for . (Note that Olver’s definition of ${\mathbf{A}}_{\nu }\left(z\right)$ omits the factor $1/\pi$ in (11.10.4).) See also Watson (1944, §10.15).

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

11.11.18		$${\mathbf{J}}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{2/3}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$
$\nu \to +\mathrm{\infty }$,
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.11.E18 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11

11.11.19		$${\mathbf{E}}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$
$\nu \to +\mathrm{\infty }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\sim$: asymptotic equality and $\nu$: real or complex order Referenced by: §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E19 Encodings: TeX, pMML, png See also: Annotations for §11.11(iii), §11.11 and Ch.11