§11.11 Asymptotic Expansions of Anger–Weber Functions

§11.11(i) Large $|z|$, Fixed $\nu$

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

 11.11.1 $\displaystyle F_{k}(\nu)$ $\displaystyle=(\nu^{2}-1^{2})(\nu^{2}-3^{2})\cdots(\nu^{2}-(2k-1)^{2}),$ $\displaystyle G_{k}(\nu)$ $\displaystyle=(\nu^{2}-2^{2})(\nu^{2}-4^{2})\cdots(\nu^{2}-(2k)^{2}).$ ⓘ Symbols: $\nu$: real or complex order, $k$: nonnegative integer, $F_{k}(\nu)$: expansion functions and $G_{k}(\nu)$: expansion functions Referenced by: 7th Erratum 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 $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta$ ($<\pi$)

 11.11.2 $\mathbf{J}_{\nu}\left(z\right)\sim J_{\nu}\left(z\right)\\ +\frac{\sin\left(\pi\nu\right)}{\pi z}\left(\sum_{k=0}^{\infty}\frac{F_{k}(\nu% )}{z^{2k}}-\frac{\nu}{z}\sum_{k=0}^{\infty}\frac{G_{k}(\nu)}{z^{2k}}\right),$
 11.11.3 $\mathbf{E}_{\nu}\left(z\right)\sim-Y_{\nu}\left(z\right)-\frac{1+\cos\left(\pi% \nu\right)}{\pi z}\sum_{k=0}^{\infty}\frac{F_{k}(\nu)}{z^{2k}}-\frac{\nu(1-% \cos\left(\pi\nu\right))}{\pi z^{2}}\sum_{k=0}^{\infty}\frac{G_{k}(\nu)}{z^{2k% }},$
 11.11.4 $\mathbf{A}_{\nu}\left(z\right)\sim\frac{1}{\pi z}\sum_{k=0}^{\infty}\frac{F_{k% }(\nu)}{z^{2k}}-\frac{\nu}{\pi z^{2}}\sum_{k=0}^{\infty}\frac{G_{k}(\nu)}{z^{2% k}}.$

§11.11(ii) Large $|\nu|$, Fixed $z$

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

 11.11.5 $\mathbf{J}_{\nu}\left(z\right)=\frac{\sin\left(\pi\nu\right)}{\pi\nu}\left(1-% \frac{\nu z}{\nu^{2}-1}+O\left(\frac{1}{\nu^{2}}\right)\right),$
 11.11.6 $\mathbf{E}_{\nu}\left(z\right)=\frac{2}{\pi\nu}\left({\sin^{2}}\left(\tfrac{1}% {2}\pi\nu\right)+\frac{\nu z}{\nu^{2}-1}{\cos^{2}}\left(\tfrac{1}{2}\pi\nu% \right)+O\left(\frac{1}{\nu^{2}}\right)\right).$

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

 11.11.7 $\displaystyle\mathbf{E}_{2n}\left(z\right)$ $\displaystyle\sim\frac{2z}{(4n^{2}-1)\pi},$ $\displaystyle\mathbf{E}_{2n+1}\left(z\right)$ $\displaystyle\sim\frac{2}{(2n+1)\pi},$ $n\to\pm\infty$.

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

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

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

where

 11.11.9 $\displaystyle a_{0}(\lambda)$ $\displaystyle=\frac{1}{1+\lambda},$ $\displaystyle a_{1}(\lambda)$ $\displaystyle=-\frac{\lambda}{2(1+\lambda)^{4}},$ $\displaystyle a_{2}(\lambda)$ $\displaystyle=\frac{9\lambda^{2}-\lambda}{24(1+\lambda)^{7}},$ $\displaystyle a_{3}(\lambda)$ $\displaystyle=-\frac{225\lambda^{3}-54\lambda^{2}+\lambda}{720(1+\lambda)^{10}}.$ ⓘ Symbols: $\lambda$: parameter and $a_{k}(\lambda)$: expansion function Referenced by: 7th Erratum 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 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim-\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(-\lambda)}{\nu^{2k+1}},$ $\nu\to+\infty$.

For fixed $\lambda$, $0<\lambda<1$,

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

where

 11.11.12 $\mu=\sqrt{1-\lambda^{2}}-\ln\left(\frac{1+\sqrt{1-\lambda^{2}}}{\lambda}\right),$ ⓘ Symbols: $\ln\NVar{z}$: 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 $\displaystyle b_{0}(\lambda)$ $\displaystyle=\frac{1}{(1-\lambda^{2})^{1/4}}$, $\displaystyle b_{1}(\lambda)$ $\displaystyle=\frac{2+3\lambda^{2}}{12(1-\lambda^{2})^{7/4}},$ $\displaystyle b_{2}(\lambda)$ $\displaystyle=\frac{4+300\lambda^{2}+81\lambda^{4}}{864(1-\lambda^{2})^{13/4}}$. ⓘ 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 $\nu\to+\infty$,

 11.11.14 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi\nu(\lambda-1)},$ $\lambda>1$,
 11.11.15 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\left(\frac{2}{\pi\nu}\right)^{1/2% }\left(\frac{1+\sqrt{1-\lambda^{2}}}{\lambda}\right)^{\nu}\frac{e^{-\nu\sqrt{1% -\lambda^{2}}}}{(1-\lambda^{2})^{1/4}},$ $0<\lambda<1$.

Also, as $\nu\to+\infty$,

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

and

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

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 $\nu\to+\infty$, 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 $\mathbf{A}_{\nu}\left(z\right)$ 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 $\mathbf{J}_{\nu}\left(\nu\right)\sim\frac{2^{1/3}}{3^{2/3}\Gamma\left(\tfrac{2% }{3}\right)\nu^{1/3}},$ $\nu\to+\infty$,
 11.11.19 $\mathbf{E}_{\nu}\left(\nu\right)\sim\frac{2^{1/3}}{3^{7/6}\Gamma\left(\tfrac{2% }{3}\right)\nu^{1/3}},$ $\nu\to+\infty$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: 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