10 Bessel Functions Bessel and Hankel Functions 10.22 Integrals 10.24 Functions of Imaginary Order

§10.23 Sums

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Permalink:: http://dlmf.nist.gov/10.23
See also:: Annotations for Ch.10

§10.23(i) Multiplication Theorem
§10.23(ii) Addition Theorems
§10.23(iii) Series Expansions of Arbitrary Functions
§10.23(iv) Compendia

§10.23(i) Multiplication Theorem

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Keywords:: Bessel functions, Hankel functions, cylinder functions, multiplication theorem, sums
Notes:: See Watson (1944, §5.22).
Permalink:: http://dlmf.nist.gov/10.23.i
See also:: Annotations for §10.23 and Ch.10

10.23.1		$\mathscr{C}_{\nu}\left(\lambda z\right)=\lambda^{\pm\nu}\sum_{k=0}^{\infty}% \frac{(\mp 1)^{k}(\lambda^{2}-1)^{k}(\tfrac{1}{2}z)^{k}}{k!}\mathscr{C}_{\nu% \pm k}\left(z\right),$
$\|\lambda^{2}-1\|<1$ .
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.74 Referenced by: §10.44(i), §10.66 Permalink: http://dlmf.nist.gov/10.23.E1 Encodings: TeX, pMML, png See also: Annotations for §10.23(i), §10.23 and Ch.10

If $\mathscr{C}=J$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary.

§10.23(ii) Addition Theorems

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Keywords:: Bessel functions, Hankel functions, addition theorems, cylinder functions, sums
Notes:: For (10.23.7) and (10.23.8) see Watson (1944, §§11.3, 11.4). (10.23.2) is obtained from (10.23.7) by taking $\chi=0$ and $\alpha=0,\pi$ .
Referenced by:: §10.60(i), §28.27, Erratum (V1.0.7) for Usability
Permalink:: http://dlmf.nist.gov/10.23.ii
Addition (effective with 1.0.7):: The cross-reference to §10.12 after (10.23.9) has been added.
Suggested 2012-12-28 by Alexander Barnett
See also:: Annotations for §10.23 and Ch.10

Neumann’s Addition Theorem

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Keywords:: Bessel functions, Neumann’s addition theorem
See also:: Annotations for §10.23(ii), §10.23 and Ch.10

10.23.2		$\mathscr{C}_{\nu}\left(u\pm v\right)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu% \mp k}\left(u\right)J_{k}\left(v\right),$
$\|v\|<\|u\|$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $k$ : nonnegative integer and $\nu$ : complex parameter A&S Ref: 9.1.75 Referenced by: §10.23(iii), §10.23(ii), §10.44(ii), §10.66 Permalink: http://dlmf.nist.gov/10.23.E2 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

The restriction $|v|<|u|$ is unnecessary when $\mathscr{C}=J$ and $\nu$ is an integer. Special cases are:

10.23.3		${J_{0}}^{2}\left(z\right)+2\sum_{k=1}^{\infty}{J_{k}}^{2}\left(z\right)=1,$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.1.76 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.23.E3 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

10.23.4		$\sum_{k=0}^{2n}(-1)^{k}J_{k}\left(z\right)J_{2n-k}\left(z\right)\\ +2\sum_{k=1}^{\infty}J_{k}\left(z\right)J_{2n+k}\left(z\right)=0,$
$n\geq 1$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.1.77 Permalink: http://dlmf.nist.gov/10.23.E4 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

10.23.5		$\sum_{k=0}^{n}J_{k}\left(z\right)J_{n-k}\left(z\right)+2\sum_{k=1}^{\infty}(-1% )^{k}J_{k}\left(z\right)J_{n+k}\left(z\right)=J_{n}\left(2z\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.1.78 Permalink: http://dlmf.nist.gov/10.23.E5 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

Graf’s and Gegenbauer’s Addition Theorems

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Keywords:: Bessel functions, Gegenbauer’s addition theorem, Graf’s addition theorem
See also:: Annotations for §10.23(ii), §10.23 and Ch.10

Define

10.23.6	$\displaystyle w$	$\displaystyle=\sqrt{u^{2}+v^{2}-2uv\cos\alpha},$
	$\displaystyle u-v\cos\alpha$	$\displaystyle=w\cos\chi,$
	$\displaystyle v\sin\alpha$	$\displaystyle=w\sin\chi,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function Permalink: http://dlmf.nist.gov/10.23.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

the branches being continuous and chosen so that $w\to u$ and $\chi\to 0$ as $v\to 0$ . If $u$ , $v$ are real and positive and $0\leq\alpha\leq\pi$ , then $w$ and $\chi$ are real and nonnegative, and the geometrical relationship is shown in Figure 10.23.1.

See accompanying text — Figure 10.23.1: Graf’s and Gegenbauer’s addition theorems. Magnify

10.23.7		$\mathscr{C}_{\nu}\left(w\right)\selection{\cos\\ \sin}(\nu\chi)=\sum_{k=-\infty}^{\infty}\mathscr{C}_{\nu+k}\left(u\right)J_{k}% \left(v\right)\selection{\cos\\ \sin}(k\alpha),$
$\|ve^{\pm i\alpha}\|<\|u\|$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer and $\nu$ : complex parameter A&S Ref: 9.1.79 Referenced by: §10.23(ii), §10.23(iii), §10.23(ii), §10.44(ii) Permalink: http://dlmf.nist.gov/10.23.E7 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

10.23.8		$\frac{\mathscr{C}_{\nu}\left(w\right)}{w^{\nu}}=2^{\nu}\Gamma\left(\nu\right)% \*\sum_{k=0}^{\infty}(\nu+k)\frac{\mathscr{C}_{\nu+k}\left(u\right)}{u^{\nu}}% \frac{J_{\nu+k}\left(v\right)}{v^{\nu}}C^{(\nu)}_{k}\left(\cos\alpha\right),$
$\nu\neq 0,-1,\dots$ , $\|ve^{\pm i\alpha}\|<\|u\|$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\cos\NVar{z}$ : cosine function, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $k$ : nonnegative integer and $\nu$ : complex parameter A&S Ref: 9.1.80 Referenced by: §10.23(ii), §10.23(ii), §10.23(ii), §10.44(ii), §10.60(i), §10.60(ii) Permalink: http://dlmf.nist.gov/10.23.E8 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

where $C^{(\nu)}_{k}\left(\cos\alpha\right)$ is Gegenbauer’s polynomial (§18.3). The restriction $|ve^{\pm i\alpha}|<|u|$ is unnecessary in (10.23.7) when $\mathscr{C}=J$ and $\nu$ is an integer, and in (10.23.8) when $\mathscr{C}=J$ .

The degenerate form of (10.23.8) when $u=\infty$ is given by

10.23.9		$e^{iv\cos\alpha}=\frac{\Gamma\left(\nu\right)}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k% =0}^{\infty}(\nu+k)i^{k}J_{\nu+k}\left(v\right)C^{(\nu)}_{k}\left(\cos\alpha% \right),$
$\nu\neq 0,-1,\dots$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $k$ : nonnegative integer and $\nu$ : complex parameter A&S Ref: 9.1.81 Referenced by: §10.23(ii) Permalink: http://dlmf.nist.gov/10.23.E9 Encodings: TeX, pMML, png See also: Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

Partial Fractions

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Keywords:: Bessel functions, Hankel functions, addition theorems, expansions in partial fractions, sums
See also:: Annotations for §10.23(ii), §10.23 and Ch.10

For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005).

§10.23(iii) Series Expansions of Arbitrary Functions

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Keywords:: Bessel functions, expansions in series of, expansions in series of Bessel functions, sums
Notes:: For (10.23.10)–(10.23.13) see Watson (1944, § 16.11). For (10.23.15)–(10.23.17) see Watson (1944, pp. 64, 67, 71, 138). For (10.23.21) see Temme (1996b, p. 247).
Referenced by:: §11.4(iv)
Permalink:: http://dlmf.nist.gov/10.23.iii
See also:: Annotations for §10.23 and Ch.10

Neumann’s Expansion

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Keywords:: Bessel functions, Neumann’s expansion, Neumann’s polynomial
See also:: Annotations for §10.23(iii), §10.23 and Ch.10

10.23.10		$f(z)=a_{0}J_{0}\left(z\right)+2\sum_{k=1}^{\infty}a_{k}J_{k}\left(z\right),$
$\|z\|<c$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $k$ : nonnegative integer, $z$ : complex variable, $a_{k}$ and $f(t)$ A&S Ref: 9.1.82 Referenced by: §10.23(iii) Permalink: http://dlmf.nist.gov/10.23.E10 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

where $c$ is the distance of the nearest singularity of the analytic function $f(z)$ from $z=0$ ,

10.23.11		$a_{k}=\frac{1}{2\pi i}\int_{\|t\|=c^{\prime}}f(t)O_{k}\left(t\right)\,\mathrm{d}t,$
$0<c^{\prime}<c$ ,
ⓘ Defines: $a_{k}$ (locally) Symbols: $O_{\NVar{n}}\left(\NVar{x}\right)$ : Neumann’s polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $k$ : nonnegative integer, $z$ : complex variable and $f(t)$ A&S Ref: 9.1.83 Referenced by: Erratum (V1.1.2) for Equation (10.23.11) Permalink: http://dlmf.nist.gov/10.23.E11 Encodings: TeX, pMML, png Errata (effective with 1.1.2): Originally the contour of integration written incorrectly as $\|z\|=c^{\prime}$ , has been corrected to be $\|t\|=c^{\prime}$ . Reported 2021-03-22 by Mark Dunster See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

and $O_{k}\left(t\right)$ is Neumann’s polynomial, defined by the generating function:

10.23.12		$\frac{1}{t-z}=J_{0}\left(z\right)O_{0}\left(t\right)+2\sum_{k=1}^{\infty}J_{k}% \left(z\right)O_{k}\left(t\right),$
$\|z\|<\|t\|$ .
ⓘ Defines: $O_{\NVar{n}}\left(\NVar{x}\right)$ : Neumann’s polynomial Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $n$ : integer, $k$ : nonnegative integer, $x$ : real variable and $z$ : complex variable A&S Ref: 9.1.84 Permalink: http://dlmf.nist.gov/10.23.E12 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

$O_{n}\left(t\right)$ is a polynomial of degree $n+1$ in $\ifrac{1}{t}:O_{0}\left(t\right)=1/t$ and

10.23.13		$O_{n}\left(t\right)=\frac{1}{4}\sum_{k=0}^{\left\lfloor n/2\right\rfloor}\frac% {(n-k-1)!n}{k!}\left(\frac{2}{t}\right)^{n-2k+1},$
$n=1,2,\dotsc$ .
ⓘ Symbols: $O_{\NVar{n}}\left(\NVar{x}\right)$ : Neumann’s polynomial, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $n$ : integer and $k$ : nonnegative integer A&S Ref: 9.1.85 Referenced by: §10.23(iii) Permalink: http://dlmf.nist.gov/10.23.E13 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

For the more general form of expansion

10.23.14		$z^{\nu}f(z)=a_{0}J_{\nu}\left(z\right)+2\sum_{k=1}^{\infty}a_{k}J_{\nu+k}\left% (z\right)$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $a_{k}$ and $f(t)$ A&S Ref: 9.1.86 Permalink: http://dlmf.nist.gov/10.23.E14 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

see Watson (1944, §16.13), and for further generalizations see Watson (1944, Chapter 16) and Erdélyi et al. (1953b, §7.10.1).

Examples

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See also:: Annotations for §10.23(iii), §10.23 and Ch.10

10.23.15		$(\tfrac{1}{2}z)^{\nu}=\sum_{k=0}^{\infty}\frac{(\nu+2k)\Gamma\left(\nu+k\right% )}{k!}J_{\nu+2k}\left(z\right),$
$\nu\neq 0,-1,-2,\dots$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.87 Referenced by: §10.22(ii), §10.23(iii), §10.44(iii), §10.74(iv) Permalink: http://dlmf.nist.gov/10.23.E15 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

10.23.16		$Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)+\gamma% \right)J_{0}\left(z\right)-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{J_{2k% }\left(z\right)}{k},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.1.89 Permalink: http://dlmf.nist.gov/10.23.E16 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

10.23.17		$Y_{n}\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(% \tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}+\frac{2}{\pi}\left(\ln\left(% \tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)J_{n}\left(z\right)-\frac{2}{% \pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)J_{n+2k}\left(z\right)}{k(n+k)},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.1.88 Referenced by: §10.23(iii), §10.44(iii) Permalink: http://dlmf.nist.gov/10.23.E17 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

where $\gamma$ is Euler’s constant and $\psi\left(n+1\right)=\Gamma'\left(n+1\right)/\Gamma\left(n+1\right)$ (§5.2).

Other examples are provided by (10.12.1)–(10.12.6), (10.23.2), and (10.23.7).

Fourier–Bessel Expansion

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Keywords:: Bessel functions, Fourier–Bessel expansion
See also:: Annotations for §10.23(iii), §10.23 and Ch.10

Assume $f(t)$ satisfies

10.23.18		$\int_{0}^{1}t^{\frac{1}{2}}\|f(t)\|\,\mathrm{d}t<\infty,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $f(t)$ Permalink: http://dlmf.nist.gov/10.23.E18 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

and define

10.23.19		$a_{m}=\frac{2}{(J_{\nu+1}\left(j_{\nu,m}\right))^{2}}\int_{0}^{1}tf(t)J_{\nu}% \left(j_{\nu,m}t\right)\,\mathrm{d}t,$
$\nu\geq-\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $m$ : integer, $\nu$ : complex parameter, $a_{k}$ and $f(t)$ Referenced by: §10.74(vii) Permalink: http://dlmf.nist.gov/10.23.E19 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

where $j_{\nu,m}$ is as in §10.21(i). If $0<x<1$ , then

10.23.20		$\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)=\sum_{m=1}^{\infty}a_{m}J_{\nu}\left(j_{% \nu,m}x\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $m$ : integer, $x$ : real variable, $\nu$ : complex parameter, $a_{k}$ and $f(t)$ Permalink: http://dlmf.nist.gov/10.23.E20 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

provided that $f(t)$ is of bounded variation (§1.4(v)) on an interval $[a,b]$ with $0<a<x<b<1$ . This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near $x=0$ and $x=1$ .

As an example,

10.23.21		$x^{\nu}=\sum_{m=1}^{\infty}\frac{2J_{\nu}\left(j_{\nu,m}x\right)}{j_{\nu,m}J_{% \nu+1}\left(j_{\nu,m}\right)},$
$\nu>0,0\leq x<1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $j_{\NVar{\nu},\NVar{m}}$ : zeros of the Bessel function $J_{\nu}\left(x\right)$ , $m$ : integer, $x$ : real variable and $\nu$ : complex parameter Referenced by: §10.23(iii) Permalink: http://dlmf.nist.gov/10.23.E21 Encodings: TeX, pMML, png See also: Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

(Note that when $x=1$ the left-hand side is 1 and the right-hand side is 0.)

Other Series Expansions

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Keywords:: Bessel functions, expansions in series of Bessel functions, sums
See also:: Annotations for §10.23(iii), §10.23 and Ch.10

For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). See also Schäfke (1960, 1961b).

§10.23(iv) Compendia

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Keywords:: Bessel functions, compendia, expansions in series of, sums
Permalink:: http://dlmf.nist.gov/10.23.iv
See also:: Annotations for §10.23 and Ch.10

For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2015, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).

10.22 Integrals 10.24 Functions of Imaginary Order

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