# §11.10 Anger–Weber Functions

## §11.10(i) Definitions

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

 11.10.1 $\mathop{\mathbf{J}_{\nu}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int_{0}^{% \pi}\mathop{\cos\/}\nolimits\!\left(\nu\theta-z\mathop{\sin\/}\nolimits\theta% \right)d\theta,$ Defines: $\mathop{\mathbf{J}_{\nu}\/}\nolimits\!\left(z\right)$: Anger function Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits z$: 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
 11.10.2 $\mathop{\mathbf{E}_{\nu}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int_{0}^{% \pi}\mathop{\sin\/}\nolimits\!\left(\nu\theta-z\mathop{\sin\/}\nolimits\theta% \right)d\theta.$ Defines: $\mathop{\mathbf{E}_{\nu}\/}\nolimits\!\left(z\right)$: Weber function Symbols: $dx$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits z$: 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

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

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

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

 11.10.4 $\mathop{\mathbf{A}_{\nu}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int_{0}^{% \infty}e^{-\nu t-z\mathop{\sinh\/}\nolimits t}dt,$ $\realpart{z}>0$. Defines: $\mathop{\mathbf{A}_{\nu}\/}\nolimits\!\left(z\right)$: Anger–Weber function Symbols: $dx$: differential of $x$, $e$: base of exponential function, $\mathop{\sinh\/}\nolimits z$: hyperbolic sine function, $\int$: integral, $\realpart{}$: real part, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.10(i), §11.11(iii) Permalink: http://dlmf.nist.gov/11.10.E4 Encodings: TeX, pMML, png

(11.10.4) also applies when $\realpart{z}=0$ and $\realpart{\nu}>0$.

## §11.10(ii) Differential Equations

The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation

 11.10.5 $\frac{{d}^{2}w}{{dz}^{2}}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{\nu^{2}}{z^{2% }}\right)w=f(\nu,z),$

where

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

or

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

## §11.10(iii) Maclaurin Series

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

where

 11.10.10 $S_{1}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(\tfrac{1}{2}z)^{2k}}{\mathop{% \Gamma\/}\nolimits\!\left(k\!+\!\tfrac{1}{2}\nu+1\right)\mathop{\Gamma\/}% \nolimits\!\left(k\!-\!\tfrac{1}{2}\nu\!+\!1\right)},$
 11.10.11 $S_{2}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(\tfrac{1}{2}z)^{2k+1}}{\mathop{% \Gamma\/}\nolimits\!\left(k\!+\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}\right)\mathop% {\Gamma\/}\nolimits\!\left(k\!-\!\tfrac{1}{2}\nu\!+\!\tfrac{3}{2}\right)}.$

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

## §11.10(v) Interrelations

 11.10.12 $\displaystyle\mathop{\mathbf{J}_{\nu}\/}\nolimits\!\left(-z\right)$ $\displaystyle=\mathop{\mathbf{J}_{-\nu}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathbf{E}_{\nu}\/}\nolimits\!\left(-z\right)$ $\displaystyle=-\mathop{\mathbf{E}_{-\nu}\/}\nolimits\!\left(z\right).$
 11.10.13 $\displaystyle\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{J}% _{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{E% }_{\nu}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{E}_{-\nu}\/}\nolimits\!% \left(z\right),$ 11.10.14 $\displaystyle\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{E}% _{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{\mathbf{J}_{-\nu}\/}\nolimits\!\left(z\right)-\mathop{% \cos\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{J}_{\nu}\/}\nolimits\!% \left(z\right).$
 11.10.15 $\mathop{\mathbf{J}_{\nu}\/}\nolimits\!\left(z\right)=\mathop{J_{\nu}\/}% \nolimits\!\left(z\right)+\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,% \mathop{\mathbf{A}_{\nu}\/}\nolimits\!\left(z\right),$
 11.10.16 $\mathop{\mathbf{E}_{\nu}\/}\nolimits\!\left(z\right)=-\mathop{Y_{\nu}\/}% \nolimits\!\left(z\right)-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\,% \mathop{\mathbf{A}_{\nu}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{A}_{-\nu}% \/}\nolimits\!\left(z\right).$

## §11.10(vi) Relations to Other Functions

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

where

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

For the Fresnel integrals $\mathop{C\/}\nolimits$ and $\mathop{S\/}\nolimits$ see §7.2(iii).

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

 11.10.22 $\mathop{\mathbf{E}_{n}\/}\nolimits\!\left(z\right)=-\mathop{\mathbf{H}_{n}\/}% \nolimits\!\left(z\right)+\frac{1}{\pi}\sum_{k=0}^{m_{1}}\frac{\mathop{\Gamma% \/}\nolimits\!\left(k+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(n% \!+\!\tfrac{1}{2}\!-\!k\right)}(\tfrac{1}{2}z)^{n-2k-1},$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(z\right)$: Struve function, $\mathop{\mathbf{E}_{\nu}\/}\nolimits\!\left(z\right)$: Weber function, $z$: 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

and

 11.10.23 $\mathop{\mathbf{E}_{-n}\/}\nolimits\!\left(z\right)=-\mathop{\mathbf{H}_{-n}\/% }\nolimits\!\left(z\right)+\frac{(-1)^{n+1}}{\pi}\sum_{k=0}^{m_{2}}\frac{% \mathop{\Gamma\/}\nolimits\!\left(n\!-\!k\!-\!\tfrac{1}{2}\right)}{\mathop{% \Gamma\/}\nolimits\!\left(k+\tfrac{3}{2}\right)}(\tfrac{1}{2}z)^{-n+2k+1},$

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

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

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

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 $\mathop{\mathbf{J}_{\nu-1}\/}\nolimits\!\left(z\right)+\mathop{\mathbf{J}_{\nu% +1}\/}\nolimits\!\left(z\right)=\frac{2\nu}{z}\mathop{\mathbf{J}_{\nu}\/}% \nolimits\!\left(z\right)-\frac{2}{\pi z}\mathop{\sin\/}\nolimits\!\left(\pi% \nu\right),$
 11.10.33 $\mathop{\mathbf{E}_{\nu-1}\/}\nolimits\!\left(z\right)+\mathop{\mathbf{E}_{\nu% +1}\/}\nolimits\!\left(z\right)=\frac{2\nu}{z}\mathop{\mathbf{E}_{\nu}\/}% \nolimits\!\left(z\right)-\frac{2}{\pi z}(1-\mathop{\cos\/}\nolimits\!\left(% \pi\nu\right)).$
 11.10.34 $\displaystyle 2\mathop{\mathbf{J}_{\nu}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{\mathbf{J}_{\nu-1}\/}\nolimits\!\left(z\right)-\mathop{% \mathbf{J}_{\nu+1}\/}\nolimits\!\left(z\right),$ 11.10.35 $\displaystyle 2\mathop{\mathbf{E}_{\nu}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\mathop{\mathbf{E}_{\nu-1}\/}\nolimits\!\left(z\right)-\mathop{% \mathbf{E}_{\nu+1}\/}\nolimits\!\left(z\right),$
 11.10.36 $z\mathop{\mathbf{J}_{\nu}\/}\nolimits'\!\left(z\right)\pm\nu\mathop{\mathbf{J}% _{\nu}\/}\nolimits\!\left(z\right)=\pm z\mathop{\mathbf{J}_{\nu\mp 1}\/}% \nolimits\!\left(z\right)\pm\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right% )}{\pi},$
 11.10.37 $z\mathop{\mathbf{E}_{\nu}\/}\nolimits'\!\left(z\right)\pm\nu\mathop{\mathbf{E}% _{\nu}\/}\nolimits\!\left(z\right)=\pm z\mathop{\mathbf{E}_{\nu\mp 1}\/}% \nolimits\!\left(z\right)\pm\frac{(1-\mathop{\cos\/}\nolimits(\pi\nu))}{\pi}.$

## §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).