11 Struve and Related Functions Related Functions 11.9 Lommel Functions 11.11 Asymptotic Expansions of Anger–Weber Functions

§11.10 Anger–Weber Functions

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Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10
See also:: Annotations for Ch.11

§11.10(i) Definitions

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Keywords:: Anger–Weber functions, analytic properties, definitions, integral representations
Notes:: See Watson (1944, pp. 308 and 312). The notation $\mathbf{A}_{\nu}\left(z\right)$ , without the factor $1/\pi$ , was introduced in Olver (1997b, p. 84).
Permalink:: http://dlmf.nist.gov/11.10.i
See also:: Annotations for §11.10 and Ch.11

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 of $x$ , $\int$ : 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 of $x$ , $\int$ : 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		$\frac{1}{\pi}\int_{0}^{2\pi}\cos\left(\nu\theta-z\sin\theta\right)\,\mathrm{d}% \theta=(1+\cos\left(2\pi\nu\right))\,\mathbf{J}_{\nu}\left(z\right)+\sin\left(% 2\pi\nu\right)\mathbf{E}_{\nu}\left(z\right).$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.10.E3 Encodings: TeX, pMML, png See also: Annotations for §11.10(i), §11.10 and Ch.11

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 of $x$ , $\mathrm{e}$ : 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

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Keywords:: Anger–Weber functions, Bessel’s equation, differential equation, inhomogeneous forms
Notes:: See Watson (1944, p. 312).
Permalink:: http://dlmf.nist.gov/11.10.ii
See also:: Annotations for §11.10 and Ch.11

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 of $f$ with respect to $x$ , $z$ : 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		$f(\nu,z)=\frac{(z-\nu)}{\pi z^{2}}\sin\left(\pi\nu\right),$
$w=\mathbf{J}_{\nu}\left(z\right)$ ,
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.10.E6 Encodings: TeX, pMML, png See also: Annotations for §11.10(ii), §11.10 and Ch.11

11.10.7		$f(\nu,z)=-\frac{1}{\pi z^{2}}(z+\nu+(z-\nu)\cos\left(\pi\nu\right)),$
$w=\mathbf{E}_{\nu}\left(z\right)$ .
ⓘ Symbols: $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.10.E7 Encodings: TeX, pMML, png See also: Annotations for §11.10(ii), §11.10 and Ch.11

§11.10(iii) Maclaurin Series

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Keywords:: Anger–Weber functions, Maclaurin series, power series, series expansions
Notes:: See Watson (1944, §10.1).
Referenced by:: §3.6(vi)
Permalink:: http://dlmf.nist.gov/11.10.iii
See also:: Annotations for §11.10 and Ch.11

11.10.8		$\mathbf{J}_{\nu}\left(z\right)=\cos\left(\tfrac{1}{2}\pi\nu\right)\,S_{1}(\nu,% z)+\sin\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,z),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.10(vi), (11.11.5) Permalink: http://dlmf.nist.gov/11.10.E8 Encodings: TeX, pMML, png See also: Annotations for §11.10(iii), §11.10 and Ch.11

11.10.9		$\mathbf{E}_{\nu}\left(z\right)=\sin\left(\tfrac{1}{2}\pi\nu\right)\,S_{1}(\nu,% z)-\cos\left(\tfrac{1}{2}\pi\nu\right)\,S_{2}(\nu,z),$
ⓘ Symbols: $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: (11.11.6) Permalink: http://dlmf.nist.gov/11.10.E9 Encodings: TeX, pMML, png See also: Annotations for §11.10(iii), §11.10 and Ch.11

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(iv) Graphics

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Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/11.10.iv
See also:: Annotations for §11.10 and Ch.11

See accompanying text — Figure 11.10.1: Anger function $\mathbf{J}_{\nu}\left(x\right)$ for $-8\leq x\leq 8$ and $\nu=0,\frac{1}{2},1,\frac{3}{2}$ . Magnify

§11.10(v) Interrelations

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Keywords:: Anger–Weber functions, interrelations
Notes:: For (11.10.12) use (11.10.1), (11.10.2). For (11.10.13)–(11.10.15) see Watson (1944, pp. 311–312). For (11.10.16) combine (11.10.14), (11.10.15), and (10.2.3).
Permalink:: http://dlmf.nist.gov/11.10.v
See also:: Annotations for §11.10 and Ch.11

11.10.12	$\displaystyle\mathbf{J}_{\nu}\left(-z\right)$	$\displaystyle=\mathbf{J}_{-\nu}\left(z\right),$
11.10.12	$\displaystyle\mathbf{E}_{\nu}\left(-z\right)$	$\displaystyle=-\mathbf{E}_{-\nu}\left(z\right).$
ⓘ 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	$\displaystyle\sin\left(\pi\nu\right)\,\mathbf{J}_{\nu}\left(z\right)$	$\displaystyle=\cos\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z\right)-\mathbf{% E}_{-\nu}\left(z\right),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order A&S Ref: 12.3.4 Referenced by: §11.10(v) Permalink: http://dlmf.nist.gov/11.10.E13 Encodings: TeX, pMML, png See also: Annotations for §11.10(v), §11.10 and Ch.11
11.10.14	$\displaystyle\sin\left(\pi\nu\right)\,\mathbf{E}_{\nu}\left(z\right)$	$\displaystyle=\mathbf{J}_{-\nu}\left(z\right)-\cos\left(\pi\nu\right)\,\mathbf% {J}_{\nu}\left(z\right).$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order A&S Ref: 12.3.5 Referenced by: §11.10(v) Permalink: http://dlmf.nist.gov/11.10.E14 Encodings: TeX, pMML, png See also: Annotations for §11.10(v), §11.10 and Ch.11

11.10.15		$\mathbf{J}_{\nu}\left(z\right)=J_{\nu}\left(z\right)+\sin\left(\pi\nu\right)\,% \mathbf{A}_{\nu}\left(z\right),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger–Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.10(v), (11.11.18), (11.11.19), (11.11.2), §11.11(iii) Permalink: http://dlmf.nist.gov/11.10.E15 Encodings: TeX, pMML, png See also: Annotations for §11.10(v), §11.10 and Ch.11

11.10.16		$\mathbf{E}_{\nu}\left(z\right)=-Y_{\nu}\left(z\right)-\cos\left(\pi\nu\right)% \,\mathbf{A}_{\nu}\left(z\right)-\mathbf{A}_{-\nu}\left(z\right).$
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger–Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.10(v), (11.11.18), (11.11.19), (11.11.3), §11.11(iii) Permalink: http://dlmf.nist.gov/11.10.E16 Encodings: TeX, pMML, png See also: Annotations for §11.10(v), §11.10 and Ch.11

§11.10(vi) Relations to Other Functions

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Keywords:: Anger–Weber functions, Fresnel integrals, Lommel functions, Struve functions, Struve functions and modified Struve functions, relation to Anger–Weber functions, relations to Anger–Weber functions, relations to other functions
Notes:: For (11.10.17), (11.10.18) see Watson (1944, pp. 308–310). For (11.10.19), (11.10.20), use (11.10.8)–(11.10.11) with $\nu=\pm\frac{1}{2}$ and identify the resulting sums with those associated with the right-hand sides via (7.6.5), (7.6.7). For (11.10.22), (11.10.23) see Watson (1944, pp. 336–337) or Erdélyi et al. (1953b, p. 40). The upper summation limit in (11.10.23) is given incorrectly in Watson (1944, p. 337), and this error is reproduced in Erdélyi et al. (1953b), as well as in later printings of Abramowitz and Stegun (1964, Chapter 12)—earlier printings of the last reference contained a different error. (11.10.23) can be derived by combining (11.2.1) with (11.10.12), (11.10.22).
Permalink:: http://dlmf.nist.gov/11.10.vi
See also:: Annotations for §11.10 and Ch.11

11.10.17	$\displaystyle\mathbf{J}_{\nu}\left(z\right)$	$\displaystyle=\frac{\sin\left(\pi\nu\right)}{\pi}(s_{{0},{\nu}}\left(z\right)-% \nu s_{{-1},{\nu}}\left(z\right)),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.10(vi) Permalink: http://dlmf.nist.gov/11.10.E17 Encodings: TeX, pMML, png See also: Annotations for §11.10(vi), §11.10 and Ch.11
11.10.18	$\displaystyle\mathbf{E}_{\nu}\left(z\right)$	$\displaystyle=-\frac{1}{\pi}(1+\cos\left(\pi\nu\right))s_{{0},{\nu}}\left(z% \right)\\ -\frac{\nu}{\pi}(1-\cos\left(\pi\nu\right))s_{{-1},{\nu}}\left(z\right).$
ⓘ Symbols: $s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$ : Lommel function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.10(vi) Permalink: http://dlmf.nist.gov/11.10.E18 Encodings: TeX, pMML, png See also: Annotations for §11.10(vi), §11.10 and Ch.11

11.10.19	$\displaystyle\mathbf{J}_{-\frac{1}{2}}\left(z\right)$	$\displaystyle=\mathbf{E}_{\frac{1}{2}}\left(z\right)=(\tfrac{1}{2}\pi z)^{-% \frac{1}{2}}(A_{+}(\chi)\cos z-A_{-}(\chi)\sin z),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $A$ : expansion function and $\chi$ Referenced by: §11.10(vi) Permalink: http://dlmf.nist.gov/11.10.E19 Encodings: TeX, pMML, png See also: Annotations for §11.10(vi), §11.10 and Ch.11
11.10.20	$\displaystyle\mathbf{J}_{\frac{1}{2}}\left(z\right)$	$\displaystyle=-\mathbf{E}_{-\frac{1}{2}}\left(z\right)=(\tfrac{1}{2}\pi z)^{-% \frac{1}{2}}(A_{+}(\chi)\sin z+A_{-}(\chi)\cos z),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $A$ : expansion function and $\chi$ Referenced by: §11.10(vi) Permalink: http://dlmf.nist.gov/11.10.E20 Encodings: TeX, pMML, png See also: Annotations for §11.10(vi), §11.10 and Ch.11

where

11.10.21	$\displaystyle A_{\pm}(\chi)$	$\displaystyle=C\left(\chi\right)\pm S\left(\chi\right),$
11.10.21	$\displaystyle\chi$	$\displaystyle=(2z/\pi)^{\frac{1}{2}}.$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable, $A$ : expansion function and $\chi$ Permalink: http://dlmf.nist.gov/11.10.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.10(vi), §11.10 and Ch.11

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	$\displaystyle m_{1}$	$\displaystyle=\left\lfloor\tfrac{1}{2}n-\tfrac{1}{2}\right\rfloor,$
11.10.24	$\displaystyle m_{2}$	$\displaystyle=\left\lceil\tfrac{1}{2}n-\tfrac{3}{2}\right\rceil.$
ⓘ Symbols: $\left\lceil\NVar{x}\right\rceil$ : ceiling of $x$ , $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ and $n$ : integer order Permalink: http://dlmf.nist.gov/11.10.E24 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.10(vi), §11.10 and Ch.11

§11.10(vii) Special Values

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Keywords:: Anger–Weber functions, special values
Notes:: For (11.10.25) use (11.10.1) and (11.10.2). For (11.10.26) use (11.10.22). (11.10.27) and (11.10.28) can be obtained by differentiation of (11.10.1) and (11.10.2), followed by straightforward manipulation of the integrals and comparison with (11.5.1) and (11.10.1). For (11.10.29) use (11.10.1) and (10.9.2).
Permalink:: http://dlmf.nist.gov/11.10.vii
See also:: Annotations for §11.10 and Ch.11

11.10.25	$\displaystyle\mathbf{J}_{\nu}\left(0\right)$	$\displaystyle=\frac{\sin\left(\pi\nu\right)}{\pi\nu},$	$\displaystyle\mathbf{E}_{\nu}\left(0\right)$	$\displaystyle=\frac{1-\cos\left(\pi\nu\right)}{\pi\nu}.$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $\nu$ : real or complex order Referenced by: §11.10(vii) Permalink: http://dlmf.nist.gov/11.10.E25 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.10(vii), §11.10 and Ch.11
11.10.26	$\displaystyle\mathbf{E}_{0}\left(z\right)$	$\displaystyle=-\mathbf{H}_{0}\left(z\right),$	$\displaystyle\mathbf{E}_{1}\left(z\right)$	$\displaystyle=\frac{2}{\pi}-\mathbf{H}_{1}\left(z\right).$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable Referenced by: §11.10(vii) Permalink: http://dlmf.nist.gov/11.10.E26 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.10(vii), §11.10 and Ch.11

11.10.27	$\displaystyle\left.\frac{\partial}{\partial\nu}\mathbf{J}_{\nu}\left(z\right)% \right\|_{\nu=0}$	$\displaystyle=\tfrac{1}{2}\pi\mathbf{H}_{0}\left(z\right),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.10(vii) Permalink: http://dlmf.nist.gov/11.10.E27 Encodings: TeX, pMML, png See also: Annotations for §11.10(vii), §11.10 and Ch.11
11.10.28	$\displaystyle\left.\frac{\partial}{\partial\nu}\mathbf{E}_{\nu}\left(z\right)% \right\|_{\nu=0}$	$\displaystyle=\tfrac{1}{2}\pi J_{0}\left(z\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.10(vii) Permalink: http://dlmf.nist.gov/11.10.E28 Encodings: TeX, pMML, png See also: Annotations for §11.10(vii), §11.10 and Ch.11

11.10.29		$\mathbf{J}_{n}\left(z\right)=J_{n}\left(z\right),$
$n\in\mathbb{Z}$ .
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\in$ : element of, $\mathbb{Z}$ : set of all integers, $z$ : complex variable and $n$ : integer order A&S Ref: 12.3.2 Referenced by: §11.10(vii), §11.11(ii) Permalink: http://dlmf.nist.gov/11.10.E29 Encodings: TeX, pMML, png See also: Annotations for §11.10(vii), §11.10 and Ch.11

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

ⓘ

Keywords:: Anger–Weber functions, products of Bessel functions, series expansions
Notes:: See Luke (1969b, p. 55).
Permalink:: http://dlmf.nist.gov/11.10.viii
See also:: Annotations for §11.10 and Ch.11

11.10.30		${\mathbf{J}_{\nu}\left(z\right)=}\\ {2\sin\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{% 2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}% \left(\tfrac{1}{2}z\right)}+{2\cos\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}% ^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z% \right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/11.10.E30 Encodings: TeX, pMML, png See also: Annotations for §11.10(viii), §11.10 and Ch.11

11.10.31		${\mathbf{E}_{\nu}\left(z\right)=}\\ {-2\cos\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}% {2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}% \left(\tfrac{1}{2}z\right)}+{2\sin\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}% ^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z% \right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/11.10.E31 Encodings: TeX, pMML, png See also: Annotations for §11.10(viii), §11.10 and Ch.11

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

§11.10(ix) Recurrence Relations and Derivatives

ⓘ

Keywords:: Anger–Weber functions, derivatives, recurrence relations
Notes:: See Watson (1944, pp. 311–312).
Permalink:: http://dlmf.nist.gov/11.10.ix
See also:: Annotations for §11.10 and Ch.11

11.10.32		$\mathbf{J}_{\nu-1}\left(z\right)+\mathbf{J}_{\nu+1}\left(z\right)=\frac{2\nu}{% z}\mathbf{J}_{\nu}\left(z\right)-\frac{2}{\pi z}\sin\left(\pi\nu\right),$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.10.E32 Encodings: TeX, pMML, png See also: Annotations for §11.10(ix), §11.10 and Ch.11

11.10.33		$\mathbf{E}_{\nu-1}\left(z\right)+\mathbf{E}_{\nu+1}\left(z\right)=\frac{2\nu}{% z}\mathbf{E}_{\nu}\left(z\right)-\frac{2}{\pi z}(1-\cos\left(\pi\nu\right)).$
ⓘ Symbols: $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.10.E33 Encodings: TeX, pMML, png See also: Annotations for §11.10(ix), §11.10 and Ch.11

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		$z\mathbf{J}_{\nu}'\left(z\right)\pm\nu\mathbf{J}_{\nu}\left(z\right)=\pm z% \mathbf{J}_{\nu\mp 1}\left(z\right)\pm\frac{\sin\left(\pi\nu\right)}{\pi},$
ⓘ Symbols: $\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.10.E36 Encodings: TeX, pMML, png See also: Annotations for §11.10(ix), §11.10 and Ch.11

11.10.37		$z\mathbf{E}_{\nu}'\left(z\right)\pm\nu\mathbf{E}_{\nu}\left(z\right)=\pm z% \mathbf{E}_{\nu\mp 1}\left(z\right)\pm\frac{(1-\cos\left(\pi\nu\right))}{\pi}.$
ⓘ Symbols: $\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Weber function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.10.E37 Encodings: TeX, pMML, png See also: Annotations for §11.10(ix), §11.10 and Ch.11

§11.10(x) Integrals and Sums

ⓘ

Keywords:: Anger–Weber functions, integrals, sums
Permalink:: http://dlmf.nist.gov/11.10.x
See also:: Annotations for §11.10 and Ch.11

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).

11.9 Lommel Functions 11.11 Asymptotic Expansions of Anger–Weber Functions

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§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

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