§11.10 Anger–Weber Functions
Contents
- §11.10(i) Definitions
- §11.10(ii) Differential Equations
- §11.10(iii) Maclaurin Series
- §11.10(iv) Graphics
- §11.10(v) Interrelations
- §11.10(vi) Relations to Other Functions
- §11.10(vii) Special Values
- §11.10(viii) Expansions in Series of Products of Bessel Functions
- §11.10(ix) Recurrence Relations and Derivatives
- §11.10(x) Integrals and Sums
§11.10(i) Definitions
The Anger function and Weber function
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,](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E1.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.](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E2.png)
Each is an entire function of and
. 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).](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E3.png)
§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),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E5.png)
where
11.10.6![f(\nu,z)=\frac{(z-\nu)}{\pi z^{2}}\mathop{\sin\/}\nolimits\!\left(\pi\nu\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E6.png)
,
![f(\nu,z)=\frac{(z-\nu)}{\pi z^{2}}\mathop{\sin\/}\nolimits\!\left(\pi\nu\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E6.png)
![w=\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/m88.png)
or
11.10.7![f(\nu,z)=-\frac{1}{\pi z^{2}}(z+\nu+(z-\nu)\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E7.png)
.
![f(\nu,z)=-\frac{1}{\pi z^{2}}(z+\nu+(z-\nu)\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E7.png)
![w=\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/m87.png)
§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),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E8.png)
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),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E9.png)
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)},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E10.png)
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)}.](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E11.png)
These expansions converge absolutely for all finite values of .
§11.10(iv) Graphics
§11.10(v) Interrelations
11.10.12
![\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(-z\right)=\mathop{\mathbf{J}_{{-\nu}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E12a.png)
![\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(-z\right)=-\mathop{\mathbf{E}_{{-\nu}}\/}\nolimits\!\left(z\right).](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E12b.png)
11.10.13
![\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{E}_{{-\nu}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E13.png)
11.10.14
![\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{\mathbf{J}_{{-\nu}}\/}\nolimits\!\left(z\right)-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right).](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E14.png)
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),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E15.png)
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).](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E16.png)
§11.10(vi) Relations to Other Functions
For ,
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}},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E22.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}},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E23.png)
where
11.10.24
![m_{1}=\left\lfloor\tfrac{1}{2}n-\tfrac{1}{2}\right\rfloor,](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E24a.png)
![m_{2}=\left\lceil\tfrac{1}{2}n-\tfrac{3}{2}\right\rceil.](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E24b.png)
§11.10(vii) Special Values
11.10.25![\displaystyle\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(0\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E25a.png)
![\displaystyle\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(0\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E25a.png)
![\displaystyle\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(0\right)=\frac{1-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu}.](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E25b.png)
11.10.26![\displaystyle\mathop{\mathbf{E}_{{0}}\/}\nolimits\!\left(z\right)=-\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E26a.png)
![\displaystyle\mathop{\mathbf{E}_{{0}}\/}\nolimits\!\left(z\right)=-\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E26a.png)
![\displaystyle\mathop{\mathbf{E}_{{1}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}-\mathop{\mathbf{H}_{{1}}\/}\nolimits\!\left(z\right).](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E26b.png)
11.10.27
![\left.\frac{\partial}{\partial\nu}\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)\right|_{{\nu=0}}=\tfrac{1}{2}\pi\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E27.png)
11.10.28
![\left.\frac{\partial}{\partial\nu}\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)\right|_{{\nu=0}}=\tfrac{1}{2}\pi\mathop{J_{{0}}\/}\nolimits\!\left(z\right).](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E28.png)
11.10.29![\mathop{\mathbf{J}_{{n}}\/}\nolimits\!\left(z\right)=\mathop{J_{{n}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E29.png)
.
![\mathop{\mathbf{J}_{{n}}\/}\nolimits\!\left(z\right)=\mathop{J_{{n}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E29.png)
![n\in\Integer](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/m86.png)
§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)},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E30.png)
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)},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E31.png)
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),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E32.png)
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)).](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E33.png)
11.10.34
![2{\mathop{\mathbf{J}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{\mathbf{J}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{J}_{{\nu+1}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E34.png)
11.10.35
![2{\mathop{\mathbf{E}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{\mathbf{E}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{E}_{{\nu+1}}\/}\nolimits\!\left(z\right),](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E35.png)
11.10.36
![z{\mathop{\mathbf{J}_{{\nu}}\/}\nolimits^{{\prime}}}\!\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},](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E36.png)
11.10.37
![z{\mathop{\mathbf{E}_{{\nu}}\/}\nolimits^{{\prime}}}\!\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}.](https://webarchive.library.unt.edu/web/20130301101223im_/http://dlmf.nist.gov/11/10/E37.png)
§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).