§11.10 Anger–Weber Functions

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

11.10.1		$${\mathbf{J}}_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\cos \left(\nu \theta -z\sin \theta \right)d\theta ,$$
ⓘ Defines: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: 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		$${\mathbf{E}}_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\pi }\sin \left(\nu \theta -z\sin \theta \right)d\theta .$$
ⓘ Defines: ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\sin z$: 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		$$\frac{1}{\pi }{\int }_0^{2\pi }\cos \left(\nu \theta -z\sin \theta \right)d\theta =(1+\cos \left(2\pi \nu \right)){\mathbf{J}}_{\nu }\left(z\right)+\sin \left(2\pi \nu \right){\mathbf{E}}_{\nu }\left(z\right).$$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: 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		$${\mathbf{A}}_{\nu }\left(z\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\nu t-z\sinh t}dt,$$
$\mathrm{\Re }z>0$.
ⓘ Defines: ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: 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 See also: Annotations for §11.10(i), §11.10 and Ch.11

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

The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation

11.10.5		$$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=f(\nu ,z),$$
ⓘ Symbols: $\frac{df}{dx}$: 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: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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

or

11.10.7		$$f(\nu ,z)=-\frac{1}{\pi z^2}(z+\nu +(z-\nu )\cos \left(\pi \nu \right)),$$
$w={\mathbf{E}}_{\nu }\left(z\right)$.
ⓘ Symbols: ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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

11.10.8		$${\mathbf{J}}_{\nu }\left(z\right)=\cos \left(\frac{1}{2}\pi \nu \right)S_1(\nu ,z)+\sin \left(\frac{1}{2}\pi \nu \right)S_2(\nu ,z),$$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.10(vi), §11.11(ii) 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		$${\mathbf{E}}_{\nu }\left(z\right)=\sin \left(\frac{1}{2}\pi \nu \right)S_1(\nu ,z)-\cos \left(\frac{1}{2}\pi \nu \right)S_2(\nu ,z),$$
ⓘ Symbols: ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order 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		$$S_1(\nu ,z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(\frac{1}{2}z)}^{2k}}{\mathrm{\Gamma }\left(k+\frac{1}{2}\nu +1\right)\mathrm{\Gamma }\left(k-\frac{1}{2}\nu +1\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer 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		$$S_2(\nu ,z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(\frac{1}{2}z)}^{2k+1}}{\mathrm{\Gamma }\left(k+\frac{1}{2}\nu +\frac{3}{2}\right)\mathrm{\Gamma }\left(k-\frac{1}{2}\nu +\frac{3}{2}\right)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer Referenced by: §11.10(vi), §11.11(ii) 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 $z$.

11.10.12		${\mathbf{J}}_{\nu }\left(-z\right)$	$={\mathbf{J}}_{-\nu }\left(z\right),$
		${\mathbf{E}}_{\nu }\left(-z\right)$	$=-{\mathbf{E}}_{-\nu }\left(z\right).$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: 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		$\sin \left(\pi \nu \right){\mathbf{J}}_{\nu }\left(z\right)$	$=\cos \left(\pi \nu \right){\mathbf{E}}_{\nu }\left(z\right)-{\mathbf{E}}_{-\nu }\left(z\right),$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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		$\sin \left(\pi \nu \right){\mathbf{E}}_{\nu }\left(z\right)$	$={\mathbf{J}}_{-\nu }\left(z\right)-\cos \left(\pi \nu \right){\mathbf{J}}_{\nu }\left(z\right).$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.10(v), §11.11(i), §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: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, ${\mathbf{A}}_{\nu }\left(z\right)$: Anger–Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.10(v), §11.11(i), §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

11.10.17		${\mathbf{J}}_{\nu }\left(z\right)$	$={\displaystyle \frac{\sin \left(\pi \nu \right)}{\pi }}(s_{0,\nu }\left(z\right)-\nu s_{-1,\nu }\left(z\right)),$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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		${\mathbf{E}}_{\nu }\left(z\right)$	$=-{\displaystyle \frac{1}{\pi }}(1+\cos \left(\pi \nu \right))s_{0,\nu }\left(z\right)-{\displaystyle \frac{\nu }{\pi }}(1-\cos \left(\pi \nu \right))s_{-1,\nu }\left(z\right).$
ⓘ Symbols: $s_{\mu ,\nu }\left(z\right)$: Lommel function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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		${\mathbf{J}}_{-\frac{1}{2}}\left(z\right)$	$={\mathbf{E}}_{\frac{1}{2}}\left(z\right)={(\frac{1}{2}\pi z)}^{-\frac{1}{2}}(A_{+}(\chi )\cos z-A_{-}(\chi )\sin z),$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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		${\mathbf{J}}_{\frac{1}{2}}\left(z\right)$	$=-{\mathbf{E}}_{-\frac{1}{2}}\left(z\right)={(\frac{1}{2}\pi z)}^{-\frac{1}{2}}(A_{+}(\chi )\sin z+A_{-}(\chi )\cos z),$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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		$A_{\pm }(\chi )$	$=C\left(\chi \right)\pm S\left(\chi \right),$
		$\chi$	$={(2z/\pi )}^{\frac{1}{2}}.$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: 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 $n=1,2,3,\mathrm{\dots }$,

11.10.22		$${\mathbf{E}}_n\left(z\right)=-{\mathbf{H}}_n\left(z\right)+\frac{1}{\pi }\sum _{k=0}^{m_1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right)}{\mathrm{\Gamma }\left(n+\frac{1}{2}-k\right)}{(\frac{1}{2}z)}^{n-2k-1},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, ${\mathbf{E}}_{\nu }\left(z\right)$: 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		$${\mathbf{E}}_{-n}\left(z\right)=-{\mathbf{H}}_{-n}\left(z\right)+\frac{{(-1)}^{n+1}}{\pi }\sum _{k=0}^{m_2}\frac{\mathrm{\Gamma }\left(n-k-\frac{1}{2}\right)}{\mathrm{\Gamma }\left(k+\frac{3}{2}\right)}{(\frac{1}{2}z)}^{-n+2k+1},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, ${\mathbf{E}}_{\nu }\left(z\right)$: 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		$m_1$	$=\lfloor \frac{1}{2}n-\frac{1}{2}\rfloor ,$
		$m_2$	$=\lceil \frac{1}{2}n-\frac{3}{2}\rceil .$
ⓘ Symbols: $\lceil x\rceil$: ceiling of $x$, $\lfloor x\rfloor$: 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

11.10.25		${\mathbf{J}}_{\nu }\left(0\right)$	$={\displaystyle \frac{\sin \left(\pi \nu \right)}{\pi \nu }},$	${\mathbf{E}}_{\nu }\left(0\right)$	$={\displaystyle \frac{1-\cos \left(\pi \nu \right)}{\pi \nu }}.$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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		${\mathbf{E}}_0\left(z\right)$	$=-{\mathbf{H}}_0\left(z\right),$	${\mathbf{E}}_1\left(z\right)$	$={\displaystyle \frac{2}{\pi }}-{\mathbf{H}}_1\left(z\right).$
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, ${\mathbf{E}}_{\nu }\left(z\right)$: 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		${{\displaystyle \frac{\partial }{\partial \nu }}{\mathbf{J}}_{\nu }\left(z\right)|}_{\nu =0}$	$=\frac{1}{2}\pi {\mathbf{H}}_0\left(z\right),$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		${{\displaystyle \frac{\partial }{\partial \nu }}{\mathbf{E}}_{\nu }\left(z\right)|}_{\nu =0}$	$=\frac{1}{2}\pi J_0\left(z\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: 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

11.10.30		$${\mathbf{J}}_{\nu }\left(z\right)=\begin{array}{l}2\sin \left(\frac{1}{2}\nu \pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{J}_{k-\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)\\ +2\cos \left(\frac{1}{2}\nu \pi \right)\underset{k=0\phantom{\prime }}{\overset{\mathrm{\infty }\phantom{\prime }}{{\sum }^{\prime }}}{(-1)}^k{J}_{k-\frac{1}{2}\nu }\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu }\left(\frac{1}{2}z\right),\end{array}$$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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		$${\mathbf{E}}_{\nu }\left(z\right)=\begin{array}{l}-2\cos \left(\frac{1}{2}\nu \pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{J}_{k-\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)\\ +2\sin \left(\frac{1}{2}\nu \pi \right)\underset{k=0\phantom{\prime }}{\overset{\mathrm{\infty }\phantom{\prime }}{{\sum }^{\prime }}}{(-1)}^k{J}_{k-\frac{1}{2}\nu }\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu }\left(\frac{1}{2}z\right),\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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.

11.10.32		$${\mathbf{J}}_{\nu -1}\left(z\right)+{\mathbf{J}}_{\nu +1}\left(z\right)=\frac{2\nu }{z}{\mathbf{J}}_{\nu }\left(z\right)-\frac{2}{\pi z}\sin \left(\pi \nu \right),$$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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: ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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		$2{\mathbf{J}}_{\nu }^{\prime }\left(z\right)$	$={\mathbf{J}}_{\nu -1}\left(z\right)-{\mathbf{J}}_{\nu +1}\left(z\right),$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: 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		$2{\mathbf{E}}_{\nu }^{\prime }\left(z\right)$	$={\mathbf{E}}_{\nu -1}\left(z\right)-{\mathbf{E}}_{\nu +1}\left(z\right),$
ⓘ Symbols: ${\mathbf{E}}_{\nu }\left(z\right)$: 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		$$z{\mathbf{J}}_{\nu }^{\prime }\left(z\right)\pm \nu {\mathbf{J}}_{\nu }\left(z\right)=\pm z{\mathbf{J}}_{\nu \mp 1}\left(z\right)\pm \frac{\sin \left(\pi \nu \right)}{\pi },$$
ⓘ Symbols: ${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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		$$z{\mathbf{E}}_{\nu }^{\prime }\left(z\right)\pm \nu {\mathbf{E}}_{\nu }\left(z\right)=\pm z{\mathbf{E}}_{\nu \mp 1}\left(z\right)\pm \frac{(1-\cos \left(\pi \nu \right))}{\pi }.$$
ⓘ Symbols: ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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

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).