§11.10 Anger–Weber Functions

Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10

§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

Notes:: See Watson (1944, pp. 308 and 312). The notation $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ , without the factor $1/\pi$ , was introduced in Olver (1997b, p. 84).
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.i

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

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E3
Encodings:: TeX, pMML, png

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

Notes:: See Watson (1944, p. 312).
Keywords:: Anger–Weber functions, Bessel’s equation
Permalink:: http://dlmf.nist.gov/11.10.ii

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

where

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

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E7
Encodings:: TeX, pMML, png

§11.10(iii) Maclaurin Series

Notes:: See Watson (1944, §10.1).
Keywords:: Anger–Weber functions, power series
Referenced by:: ¶ ‣ §3.6(vi)
Permalink:: http://dlmf.nist.gov/11.10.iii

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

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E9
Encodings:: TeX, pMML, png

where

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

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

§11.10(iv) Graphics

Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/11.10.iv

Figure 11.10.1: Anger function $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ for $-8\leq x\leq 8$ and $\nu=0,\frac{1}{2},1,\frac{3}{2}$ .

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $x$ : real variable and $\nu$ : real or complex order
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.F1
Encodings:: pdf, png

Figure 11.10.2: Weber function $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(x\right)$ for $-8\leq x\leq 8$ and $\nu=0,\frac{1}{2},1,\frac{3}{2}$ .

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $x$ : real variable and $\nu$ : real or complex order
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.F2
Encodings:: pdf, png

Visualization Help

Figure 11.10.3: Anger function $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ for $-10\leq x\leq 10$ and $0\leq\nu\leq 5$ .

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $x$ : real variable and $\nu$ : real or complex order
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.F3
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 11.10.4: Weber function $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(x\right)$ for $-10\leq x\leq 10$ and $0\leq\nu\leq 5$ .

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $x$ : real variable and $\nu$ : real or complex order
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.F4
Encodings:: VRML, X3D, pdf, png

§11.10(v) 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).
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.v

11.10.12

$\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(-z\right)=\mathop{\mathbf{J}_{{-\nu}}\/}\nolimits\!\left(z\right),$

$\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(-z\right)=-\mathop{\mathbf{E}_{{-\nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

11.10.13 $\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{E}_{{-\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

11.10.14 $\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{\mathbf{J}_{{-\nu}}\/}\nolimits\!\left(z\right)-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

11.10.15 $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)+\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, $\mathop{\sin\/}\nolimits z$ : 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

11.10.16 $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)=-\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)\,\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{A}_{{-\nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\mathbf{A}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger–Weber function, $\mathop{\cos\/}\nolimits z$ : 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

§11.10(vi) 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).
Keywords:: Anger–Weber functions, Fresnel integrals, Lommel functions, Struve functions, Struve functions and modified Struve functions
Permalink:: http://dlmf.nist.gov/11.10.vi

11.10.17 $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}{\pi}(\mathop{s_{{{0},{\nu}}}\/}\nolimits\!\left(z\right)-\nu\mathop{s_{{{-1},{\nu}}}\/}\nolimits\!\left(z\right)),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $\mathop{\sin\/}\nolimits z$ : 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

11.10.18 $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)=-\frac{1}{\pi}(1+\mathop{\cos\/}\nolimits\!\left(\pi\nu\right))\mathop{s_{{{0},{\nu}}}\/}\nolimits\!\left(z\right)\\ -\frac{\nu}{\pi}(1-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right))\mathop{s_{{{-1},{\nu}}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$ : Lommel function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : 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

11.10.19 $\mathop{\mathbf{J}_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\mathop{\mathbf{E}_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{{-\frac{1}{2}}}(A_{{+}}(\chi)\mathop{\cos\/}\nolimits z-A_{{-}}(\chi)\mathop{\sin\/}\nolimits z),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

11.10.20 $\mathop{\mathbf{J}_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)=-\mathop{\mathbf{E}_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)\\ =(\tfrac{1}{2}\pi z)^{{-\frac{1}{2}}}(A_{{+}}(\chi)\mathop{\sin\/}\nolimits z+A_{{-}}(\chi)\mathop{\cos\/}\nolimits z),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

where

11.10.21

$A_{{\pm}}(\chi)=\mathop{C\/}\nolimits\!\left(\chi\right)\pm\mathop{S\/}\nolimits\!\left(\chi\right),$

$\chi=(2z/\pi)^{{\frac{1}{2}}}.$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $z$ : complex variable, $A$ : expansion function and $\chi$
Permalink:: http://dlmf.nist.gov/11.10.E21
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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:: 12.3.7 (with upper limit corrected.)
Referenced by:: §11.10(vi), Ch.11
Permalink:: http://dlmf.nist.gov/11.10.E23
Encodings:: TeX, pMML, png

where

11.10.24

$m_{1}=\left\lfloor\tfrac{1}{2}n-\tfrac{1}{2}\right\rfloor,$

$m_{2}=\left\lceil\tfrac{1}{2}n-\tfrac{3}{2}\right\rceil.$

Symbols:: $\left\lceil x\right\rceil$ : ceiling of $x$ , $\left\lfloor x\right\rfloor$ : floor of $x$ and $n$ : integer order
Permalink:: http://dlmf.nist.gov/11.10.E24
Encodings:: TeX, TeX, pMML, pMML, png, png

§11.10(vii) 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).
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.vii

11.10.25 $\displaystyle \mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(0\right) =\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu},$ $\displaystyle \mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(0\right) =\frac{1-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu}.$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

11.10.26 $\displaystyle \mathop{\mathbf{E}_{{0}}\/}\nolimits\!\left(z\right) =-\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right),$ $\displaystyle \mathop{\mathbf{E}_{{1}}\/}\nolimits\!\left(z\right) =\frac{2}{\pi}-\mathop{\mathbf{H}_{{1}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function and $z$ : complex variable
Referenced by:: §11.10(vii)
Permalink:: http://dlmf.nist.gov/11.10.E26
Encodings:: TeX, TeX, pMML, pMML, png, png

11.10.27 $\left.\frac{\partial}{\partial\nu}\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)\right|_{{\nu=0}}=\tfrac{1}{2}\pi\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : 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

11.10.28 $\left.\frac{\partial}{\partial\nu}\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)\right|_{{\nu=0}}=\tfrac{1}{2}\pi\mathop{J_{{0}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : 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

11.10.29 $\mathop{\mathbf{J}_{{n}}\/}\nolimits\!\left(z\right)=\mathop{J_{{n}}\/}\nolimits\!\left(z\right),$ $n\in\Integer$ .

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\in$ : element of, $\Integer$ : 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

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

Notes:: See Luke (1969b, p. 55).
Keywords:: Anger–Weber functions
Permalink:: http://dlmf.nist.gov/11.10.viii

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

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

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

§11.10(ix) Recurrence Relations and Derivatives

Notes:: See Watson (1944, pp. 311–312).
Keywords:: Anger–Weber functions, derivatives
Permalink:: http://dlmf.nist.gov/11.10.ix

11.10.32 $\mathop{\mathbf{J}_{{\nu-1}}\/}\nolimits\!\left(z\right)+\mathop{\mathbf{J}_{{\nu+1}}\/}\nolimits\!\left(z\right)=\frac{2\nu}{z}\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)-\frac{2}{\pi z}\mathop{\sin\/}\nolimits\!\left(\pi\nu\right),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E32
Encodings:: TeX, pMML, png

11.10.33 $\mathop{\mathbf{E}_{{\nu-1}}\/}\nolimits\!\left(z\right)+\mathop{\mathbf{E}_{{\nu+1}}\/}\nolimits\!\left(z\right)=\frac{2\nu}{z}\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)-\frac{2}{\pi z}(1-\mathop{\cos\/}\nolimits\!\left(\pi\nu\right)).$

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E33
Encodings:: TeX, pMML, png

11.10.34 $2{\mathop{\mathbf{J}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{\mathbf{J}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{J}_{{\nu+1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E34
Encodings:: TeX, pMML, png

11.10.35 $2{\mathop{\mathbf{E}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{\mathbf{E}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{E}_{{\nu+1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E35
Encodings:: TeX, pMML, png

11.10.36 $z{\mathop{\mathbf{J}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)\pm\nu\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)=\pm z\mathop{\mathbf{J}_{{\nu\mp 1}}\/}\nolimits\!\left(z\right)\pm\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}{\pi},$

Symbols:: $\mathop{\mathbf{J}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Anger function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E36
Encodings:: TeX, pMML, png

11.10.37 $z{\mathop{\mathbf{E}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)\pm\nu\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)=\pm z\mathop{\mathbf{E}_{{\nu\mp 1}}\/}\nolimits\!\left(z\right)\pm\frac{(1-\mathop{\cos\/}\nolimits(\pi\nu))}{\pi}.$

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $\mathop{\cos\/}\nolimits z$ : cosine function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.10.E37
Encodings:: TeX, pMML, png

§11.10(x) Integrals and Sums

Keywords:: Anger–Weber functions, integrals
Permalink:: http://dlmf.nist.gov/11.10.x

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).

§11.10 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