# §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 $\displaystyle F_{k}(\nu)$ $\displaystyle=(\nu^{2}-1^{2})(\nu^{2}-3^{2})\cdots(\nu^{2}-(2k-1)^{2})=(-4)^{k% }{\left(\frac{1}{2}-\frac{\nu}{2}\right)_{k}}{\left(\frac{1}{2}+\frac{\nu}{2}% \right)_{k}},$ $\displaystyle G_{k}(\nu)$ $\displaystyle=(\nu^{2}-2^{2})(\nu^{2}-4^{2})\cdots(\nu^{2}-(2k)^{2})=(-4)^{k}{% \left(1-\frac{\nu}{2}\right)_{k}}{\left(1+\frac{\nu}{2}\right)_{k}}.$ ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\nu$: 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, Erratum (V1.1.2) for Equation (11.11.1) Permalink: http://dlmf.nist.gov/11.11.E1 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.1.2): Pochhammer symbol representations for $F_{k}(\nu)$ and $G_{k}(\nu)$ were inserted. 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 $|\operatorname{ph}z|\leq\pi-\delta$

 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 $\mathbf{A}_{\nu}\left(z\right)\sim\frac{1}{\pi z}\sum_{k=0}^{\infty}\frac{F_{k% }(\nu)}{z^{2k}}-\frac{\nu}{\pi z^{2}}\sum_{k=0}^{\infty}\frac{G_{k}(\nu)}{z^{2% k}}.$

For sharp error bounds and exponentially-improved extensions, see Nemes (2018).

## §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 $\displaystyle\mathbf{E}_{2n}\left(z\right)$ $\displaystyle\sim\frac{2z}{(4n^{2}-1)\pi},$ $\displaystyle\mathbf{E}_{2n+1}\left(z\right)$ $\displaystyle\sim\frac{2}{(2n+1)\pi},$ ⓘ Symbols: $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $n$: integer order Proof sketch: Derivable using (11.6.5). 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

as $n\to\pm\infty$.

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

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

 11.11.8 $\mathbf{A}_{\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$, ⓘ Symbols: $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\operatorname{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_{k}(\lambda)$: expansion function Sources: Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c) Referenced by: (11.11.18), (11.11.19), §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E8 Encodings: TeX, pMML, png Clarification (effective with 1.1.2): The constraint which originally was $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta(<\pi)$, has been replaced to be $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$. Suggested 2021-04-05 by Gergő Nemes See also: Annotations for §11.11(iii), §11.11 and Ch.11

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: 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

In general,

 11.11.9_5 $a_{k+1}(\lambda)=\frac{\lambda}{1-\lambda^{2}}\frac{\lambda a^{\prime\prime}_{% k}(\lambda)+a^{\prime}_{k}(\lambda)}{(2k+1)(2k+2)},$ $k=0,1,2,\ldots$. ⓘ Symbols: $k$: nonnegative integer, $\lambda$: parameter and $a_{k}(\lambda)$: expansion function Referenced by: Erratum (V1.1.2) for Additions Permalink: http://dlmf.nist.gov/11.11.E9_5 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §11.11(iii), §11.11 and Ch.11

For fixed $\lambda(>1)$,

 11.11.10 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim-\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(-\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$. ⓘ Symbols: $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\operatorname{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_{k}(\lambda)$: expansion function Sources: Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c) Proof sketch: Apply Laplace’s method to the integral (11.10.4). Referenced by: (11.11.18), (11.11.19), §11.11(iii), §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E10 Encodings: TeX, pMML, png Clarification (effective with 1.1.2): The constraint which was originally $\nu\to+\infty$, has been extended to be $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$. Suggested 2021-04-05 by Gergő Nemes See also: Annotations for §11.11(iii), §11.11 and Ch.11

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

 11.11.11 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\left(\frac{2}{\pi\nu}\right)^{1/2% }e^{-\nu\mu}\sum_{k=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{k}}b_{k}(% \lambda)}{\nu^{k}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$, ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: 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, $\operatorname{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $b_{k}(\lambda)$: expansion function Sources: Dingle (1973, p. 388); Olver (1997b, pp. 103 and 352) Proof sketch: Derivable by applying Laplace’s method to the integral (11.10.4). Permalink: http://dlmf.nist.gov/11.11.E11 Encodings: TeX, pMML, png Clarification (effective with 1.1.2): The constraint which was originally $\nu\to+\infty$, has been extended to be $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$. Suggested 2021-04-05 by Gergő Nemes 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: $\ln\NVar{z}$: 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 general,

 11.11.13_5 ${\left(\tfrac{1}{2}\right)_{k}}b_{k}(\lambda)=\frac{(-1)^{k}}{\left(1-\lambda^% {2}\right)^{1/4}}U_{k}\left(\frac{1}{\sqrt{1-\lambda^{2}}}\right),$ $k=0,1,2,\ldots$, ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $k$: nonnegative integer, $\lambda$: parameter and $b_{k}(\lambda)$: expansion function Referenced by: Erratum (V1.1.2) for Additions Permalink: http://dlmf.nist.gov/11.11.E13_5 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §11.11(iii), §11.11 and Ch.11

with the $U_{k}$ defined in §10.41(ii).

In particular, as $\nu\to\infty$,

 11.11.14 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi\nu(\lambda-1)},$ $\lambda>1$, $|\operatorname{ph}\nu|\leq\pi-\delta$, ⓘ Symbols: $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter Source: Olver (1997b, pp. 103 and 352) Referenced by: §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E14 Encodings: TeX, pMML, png Clarification (effective with 1.1.2): The constraint has been extended to include $|\operatorname{ph}\nu|\leq\pi-\delta$. Suggested 2021-04-06 by Gergő Nemes See also: Annotations for §11.11(iii), §11.11 and Ch.11
 11.11.15 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\left(\frac{2}{\pi\nu}\right)^{1/2% }\left(\frac{1+\sqrt{1-\lambda^{2}}}{\lambda}\right)^{\nu}\frac{e^{-\nu\sqrt{1% -\lambda^{2}}}}{(1-\lambda^{2})^{1/4}},$ $0<\lambda<1$, $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$. ⓘ Symbols: $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\operatorname{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter Source: Olver (1997b, pp. 103 and 352) Permalink: http://dlmf.nist.gov/11.11.E15 Encodings: TeX, pMML, png Clarification (effective with 1.1.2): The constraint has been extended to include $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$. Suggested 2021-04-06 by Gergő Nemes See also: Annotations for §11.11(iii), §11.11 and Ch.11

Also, as $\nu\to\infty$ in $|\operatorname{ph}\nu|\leq 2\pi-\delta$,

 11.11.16 $\mathbf{A}_{-\nu}\left(\nu\right)\sim\frac{2^{4/3}}{3^{7/6}\Gamma\left(\tfrac{% 2}{3}\right)\nu^{1/3}},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger–Weber function, $\sim$: asymptotic equality and $\nu$: real or complex order Sources: Olver (1997b, pp. 103 and 352); Nemes (2020) Referenced by: (11.11.18), (11.11.19), §11.11(iii), §11.11(iii) 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 $\mathbf{A}_{-\nu}\left(\nu+a\nu^{1/3}\right)=2^{1/3}\nu^{-1/3}\operatorname{Hi% }\left(-2^{1/3}a\right)+O\left(\nu^{-1}\right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\operatorname{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger–Weber function, $\nu$: real or complex order and $a_{k}(\lambda)$: expansion function Sources: Olver (1997b, pp. 103 and 352); Nemes (2020) Referenced by: §11.11(iii), §11.11(iii), §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 complex values of $a$. For the Scorer function $\operatorname{Hi}$ see §9.12(i).

Error bounds for (11.11.8) and (11.11.10) are given in Meijer (1932) and Nemes (2014b, c). The later references also contain exponentially-improved extensions of (11.11.8) and (11.11.10). For an extension of (11.11.17) (and (11.11.16)) into a complete asymptotic expansion, see Nemes (2020).

When $\nu$ is real and positive, all of (11.11.10)–(11.11.17) can be regarded as special cases of two asymptotic expansions given in Olver (1997b, pp. 352–360) for $\mathbf{A}_{-\nu}\left(\lambda\nu\right)$ as $\nu\to+\infty$, one being uniform for $0<\lambda\leq 1$, and the other being uniform for $\lambda\geq 1$. (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 $\mathbf{E}_{\nu}\left(\lambda\nu\right)$, with $0<\lambda<1$ or $\lambda>1$, 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). Furthermore,

 11.11.18 $\mathbf{J}_{\nu}\left(\nu\right)\sim\frac{2^{1/3}}{3^{2/3}\Gamma\left(\tfrac{2% }{3}\right)\nu^{1/3}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$, ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger function, $\Gamma\left(\NVar{z}\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant Source: Olver (1997b, pp. 103 and 352) Proof sketch: Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42). Referenced by: §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E18 Encodings: TeX, pMML, png Clarification (effective with 1.1.2): The constraint which originally was $\nu\to+\infty$, has been extended to be $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$. Suggested 2021-04-05 by Gergő Nemes See also: Annotations for §11.11(iii), §11.11 and Ch.11
 11.11.19 $\mathbf{E}_{\nu}\left(\nu\right)\sim\frac{2^{1/3}}{3^{7/6}\Gamma\left(\tfrac{2% }{3}\right)\nu^{1/3}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant Source: Olver (1997b, pp. 103 and 352) Proof sketch: Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42). Referenced by: §11.11(iii) Permalink: http://dlmf.nist.gov/11.11.E19 Encodings: TeX, pMML, png Clarification (effective with 1.1.2): The constraint which originally was $\nu\to+\infty$, has been extended to be $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$. Suggested 2021-04-05 by Gergő Nemes See also: Annotations for §11.11(iii), §11.11 and Ch.11