# §11.10 Anger–Weber Functions

## §11.10(i) Definitions

The Anger function $\mathbf{J}_{\nu}\left(z\right)$ and Weber function $\mathbf{E}_{\nu}\left(z\right)$ are defined by

 11.10.1 $\mathbf{J}_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(\nu\theta-% z\sin\theta\right)\mathrm{d}\theta,$ ⓘ Defines: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\sin\NVar{z}$: sine function, $z$: complex variable and $\nu$: real or complex order A&S Ref: 12.3.1 Referenced by: §11.10(v), §11.10(vii) Permalink: http://dlmf.nist.gov/11.10.E1 Encodings: TeX, pMML, png See also: Annotations for §11.10(i), §11.10 and Ch.11
 11.10.2 $\mathbf{E}_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\sin\left(\nu\theta-% z\sin\theta\right)\mathrm{d}\theta.$ ⓘ Defines: $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: Weber function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\sin\NVar{z}$: sine function, $z$: complex variable and $\nu$: real or complex order A&S Ref: 12.3.2 Referenced by: §11.10(v), §11.10(vii) Permalink: http://dlmf.nist.gov/11.10.E2 Encodings: TeX, pMML, png See also: Annotations for §11.10(i), §11.10 and Ch.11

Each is an entire function of $z$ and $\nu$. Also,

 11.10.3 $\frac{1}{\pi}\int_{0}^{2\pi}\cos\left(\nu\theta-z\sin\theta\right)\mathrm{d}% \theta=(1+\cos\left(2\pi\nu\right))\,\mathbf{J}_{\nu}\left(z\right)+\sin\left(% 2\pi\nu\right)\mathbf{E}_{\nu}\left(z\right).$

The associated Anger–Weber function $\mathbf{A}_{\nu}\left(z\right)$ is defined by

 11.10.4 $\mathbf{A}_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\infty}e^{-\nu t-z\sinh t% }\mathrm{d}t,$ $\Re z>0$. ⓘ Defines: $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger–Weber function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\sinh\NVar{z}$: hyperbolic sine function, $\int$: integral, $\Re$: real part, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.10(i), (11.11.10), (11.11.11), §11.11(iii) Permalink: http://dlmf.nist.gov/11.10.E4 Encodings: TeX, pMML, png See also: Annotations for §11.10(i), §11.10 and Ch.11

(11.10.4) also applies when $\Re z=0$ and $\Re\nu>0$.

## §11.10(ii) Differential Equations

The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation

 11.10.5 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{% \mathrm{d}z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w=f(\nu,z),$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $z$: complex variable and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.10.E5 Encodings: TeX, pMML, png See also: Annotations for §11.10(ii), §11.10 and Ch.11

where

 11.10.6 $f(\nu,z)=\frac{(z-\nu)}{\pi z^{2}}\sin\left(\pi\nu\right),$ $w=\mathbf{J}_{\nu}\left(z\right)$,

or

 11.10.7 $f(\nu,z)=-\frac{1}{\pi z^{2}}(z+\nu+(z-\nu)\cos\left(\pi\nu\right)),$ $w=\mathbf{E}_{\nu}\left(z\right)$.

## §11.10(iii) Maclaurin Series

 11.10.8 $\mathbf{J}_{\nu}\left(z\right)=\cos\left(\tfrac{1}{2}\pi\nu\right)\,S_{1}(\nu,% z)+\sin\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,z),$
 11.10.9 $\mathbf{E}_{\nu}\left(z\right)=\sin\left(\tfrac{1}{2}\pi\nu\right)\,S_{1}(\nu,% z)-\cos\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,z),$

where

 11.10.10 $S_{1}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(\tfrac{1}{2}z)^{2k}}{\Gamma% \left(k\!+\!\tfrac{1}{2}\nu+1\right)\Gamma\left(k\!-\!\tfrac{1}{2}\nu\!+\!1% \right)},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer Referenced by: (11.11.5), (11.11.6) Permalink: http://dlmf.nist.gov/11.10.E10 Encodings: TeX, pMML, png See also: Annotations for §11.10(iii), §11.10 and Ch.11
 11.10.11 $S_{2}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(\tfrac{1}{2}z)^{2k+1}}{\Gamma% \left(k\!+\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}\right)\Gamma\left(k\!-\!\tfrac{1}% {2}\nu\!+\!\tfrac{3}{2}\right)}.$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer Referenced by: §11.10(vi), (11.11.5), (11.11.6) Permalink: http://dlmf.nist.gov/11.10.E11 Encodings: TeX, pMML, png See also: Annotations for §11.10(iii), §11.10 and Ch.11

These expansions converge absolutely for all finite values of $z$.

## §11.10(v) Interrelations

 11.10.12 $\displaystyle\mathbf{J}_{\nu}\left(-z\right)$ $\displaystyle=\mathbf{J}_{-\nu}\left(z\right),$ $\displaystyle\mathbf{E}_{\nu}\left(-z\right)$ $\displaystyle=-\mathbf{E}_{-\nu}\left(z\right).$ ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: Weber function, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.10(v), §11.10(vi) Permalink: http://dlmf.nist.gov/11.10.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.10(v), §11.10 and Ch.11
 11.10.13 $\displaystyle\sin\left(\pi\nu\right)\,\mathbf{J}_{\nu}\left(z\right)$ $\displaystyle=\cos\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z\right)-\mathbf{% E}_{-\nu}\left(z\right),$ 11.10.14 $\displaystyle\sin\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z\right)$ $\displaystyle=\mathbf{J}_{-\nu}\left(z\right)-\cos\left(\pi\nu\right)\,\mathbf% {J}_{\nu}\left(z\right).$
 11.10.15 $\mathbf{J}_{\nu}\left(z\right)=J_{\nu}\left(z\right)+\sin\left(\pi\nu\right)\,% \mathbf{A}_{\nu}\left(z\right),$
 11.10.16 $\mathbf{E}_{\nu}\left(z\right)=-Y_{\nu}\left(z\right)-\cos\left(\pi\nu\right)% \,\mathbf{A}_{\nu}\left(z\right)-\mathbf{A}_{-\nu}\left(z\right).$

## §11.10(vi) Relations to Other Functions

 11.10.17 $\displaystyle\mathbf{J}_{\nu}\left(z\right)$ $\displaystyle=\frac{\sin\left(\pi\nu\right)}{\pi}(s_{{0},{\nu}}\left(z\right)-% \nu s_{{-1},{\nu}}\left(z\right)),$ 11.10.18 $\displaystyle\mathbf{E}_{\nu}\left(z\right)$ $\displaystyle=-\frac{1}{\pi}(1+\cos\left(\pi\nu\right))s_{{0},{\nu}}\left(z% \right)\\ -\frac{\nu}{\pi}(1-\cos\left(\pi\nu\right))s_{{-1},{\nu}}\left(z\right).$
 11.10.19 $\displaystyle\mathbf{J}_{-\frac{1}{2}}\left(z\right)$ $\displaystyle=\mathbf{E}_{\frac{1}{2}}\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\cos z-A_{-}(\chi)\sin z),$ 11.10.20 $\displaystyle\mathbf{J}_{\frac{1}{2}}\left(z\right)$ $\displaystyle=-\mathbf{E}_{-\frac{1}{2}}\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\sin z+A_{-}(\chi)\cos z),$

where

 11.10.21 $\displaystyle A_{\pm}(\chi)$ $\displaystyle=C\left(\chi\right)\pm S\left(\chi\right),$ $\displaystyle\chi$ $\displaystyle=(2z/\pi)^{\frac{1}{2}}.$

For the Fresnel integrals $C$ and $S$ see §7.2(iii).

For $n=1,2,3,\dots$,

 11.10.22 $\mathbf{E}_{n}\left(z\right)=-\mathbf{H}_{n}\left(z\right)+\frac{1}{\pi}\sum_{% k=0}^{m_{1}}\frac{\Gamma\left(k+\tfrac{1}{2}\right)}{\Gamma\left(n\!+\!\tfrac{% 1}{2}\!-\!k\right)}(\tfrac{1}{2}z)^{n-2k-1},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$: Struve function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer A&S Ref: 12.3.6 (with upper limit corrected in later printings.) Referenced by: §11.10(vi), §11.10(vii) Permalink: http://dlmf.nist.gov/11.10.E22 Encodings: TeX, pMML, png See also: Annotations for §11.10(vi), §11.10 and Ch.11

and

 11.10.23 $\mathbf{E}_{-n}\left(z\right)=-\mathbf{H}_{-n}\left(z\right)+\frac{(-1)^{n+1}}% {\pi}\sum_{k=0}^{m_{2}}\frac{\Gamma\left(n\!-\!k\!-\!\tfrac{1}{2}\right)}{% \Gamma\left(k+\tfrac{3}{2}\right)}(\tfrac{1}{2}z)^{-n+2k+1},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$: Struve function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: integer order and $k$: nonnegative integer A&S Ref: 12.3.7 (with upper limit corrected.) Referenced by: §11.10(vi), Ch.11 Permalink: http://dlmf.nist.gov/11.10.E23 Encodings: TeX, pMML, png See also: Annotations for §11.10(vi), §11.10 and Ch.11

where

 11.10.24 $\displaystyle m_{1}$ $\displaystyle=\left\lfloor\tfrac{1}{2}n-\tfrac{1}{2}\right\rfloor,$ $\displaystyle m_{2}$ $\displaystyle=\left\lceil\tfrac{1}{2}n-\tfrac{3}{2}\right\rceil.$

## §11.10(vii) Special Values

 11.10.25 $\displaystyle\mathbf{J}_{\nu}\left(0\right)$ $\displaystyle=\frac{\sin\left(\pi\nu\right)}{\pi\nu},$ $\displaystyle\mathbf{E}_{\nu}\left(0\right)$ $\displaystyle=\frac{1-\cos\left(\pi\nu\right)}{\pi\nu}.$ 11.10.26 $\displaystyle\mathbf{E}_{0}\left(z\right)$ $\displaystyle=-\mathbf{H}_{0}\left(z\right),$ $\displaystyle\mathbf{E}_{1}\left(z\right)$ $\displaystyle=\frac{2}{\pi}-\mathbf{H}_{1}\left(z\right).$
 11.10.27 $\displaystyle\left.\frac{\partial}{\partial\nu}\mathbf{J}_{\nu}\left(z\right)% \right|_{\nu=0}$ $\displaystyle=\tfrac{1}{2}\pi\mathbf{H}_{0}\left(z\right),$ 11.10.28 $\displaystyle\left.\frac{\partial}{\partial\nu}\mathbf{E}_{\nu}\left(z\right)% \right|_{\nu=0}$ $\displaystyle=\tfrac{1}{2}\pi J_{0}\left(z\right).$
 11.10.29 $\mathbf{J}_{n}\left(z\right)=J_{n}\left(z\right),$ $n\in\mathbb{Z}$.

## §11.10(viii) Expansions in Series of Products of Bessel Functions

 11.10.30 ${\mathbf{J}_{\nu}\left(z\right)=}\\ {2\sin\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{% 2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}% \left(\tfrac{1}{2}z\right)}+{2\cos\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}% ^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z% \right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},$
 11.10.31 ${\mathbf{E}_{\nu}\left(z\right)=}\\ {-2\cos\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}% {2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}% \left(\tfrac{1}{2}z\right)}+{2\sin\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}% ^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z% \right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},$

where the prime on the second summation symbols means that the first term is to be halved.

## §11.10(ix) Recurrence Relations and Derivatives

 11.10.32 $\mathbf{J}_{\nu-1}\left(z\right)+\mathbf{J}_{\nu+1}\left(z\right)=\frac{2\nu}{% z}\mathbf{J}_{\nu}\left(z\right)-\frac{2}{\pi z}\sin\left(\pi\nu\right),$
 11.10.33 $\mathbf{E}_{\nu-1}\left(z\right)+\mathbf{E}_{\nu+1}\left(z\right)=\frac{2\nu}{% z}\mathbf{E}_{\nu}\left(z\right)-\frac{2}{\pi z}(1-\cos\left(\pi\nu\right)).$
 11.10.34 $\displaystyle 2\mathbf{J}_{\nu}'\left(z\right)$ $\displaystyle=\mathbf{J}_{\nu-1}\left(z\right)-\mathbf{J}_{\nu+1}\left(z\right),$ ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$: Anger function, $z$: complex variable and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.10.E34 Encodings: TeX, pMML, png See also: Annotations for §11.10(ix), §11.10 and Ch.11 11.10.35 $\displaystyle 2\mathbf{E}_{\nu}'\left(z\right)$ $\displaystyle=\mathbf{E}_{\nu-1}\left(z\right)-\mathbf{E}_{\nu+1}\left(z\right),$ ⓘ Symbols: $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$: Weber function, $z$: complex variable and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.10.E35 Encodings: TeX, pMML, png See also: Annotations for §11.10(ix), §11.10 and Ch.11
 11.10.36 $z\mathbf{J}_{\nu}'\left(z\right)\pm\nu\mathbf{J}_{\nu}\left(z\right)=\pm z% \mathbf{J}_{\nu\mp 1}\left(z\right)\pm\frac{\sin\left(\pi\nu\right)}{\pi},$
 11.10.37 $z\mathbf{E}_{\nu}'\left(z\right)\pm\nu\mathbf{E}_{\nu}\left(z\right)=\pm z% \mathbf{E}_{\nu\mp 1}\left(z\right)\pm\frac{(1-\cos\left(\pi\nu\right))}{\pi}.$

## §11.10(x) Integrals and Sums

For collections of integral representations and integrals see Erdélyi et al. (1954a, §§4.19 and 5.17), Marichev (1983, pp. 194–195 and 214–215), Oberhettinger (1972, p. 128), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105 and 189–190), Prudnikov et al. (1990, §§1.5 and 2.8), Prudnikov et al. (1992a, §3.18), Prudnikov et al. (1992b, §3.18), and Zanovello (1977).

For sums see Hansen (1975, pp. 456–457) and Prudnikov et al. (1990, §§6.4.2–6.4.3).