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

§10.23 Sums

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§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

Notes:: See Watson (1944, §5.22).
Keywords:: Bessel functions, Hankel functions, cylinder functions
Permalink:: http://dlmf.nist.gov/10.23.i

10.23.1 $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda z\right)=\lambda^{{\pm% \nu}}\sum_{{k=0}}^{\infty}\frac{(\mp 1)^{k}(\lambda^{2}-1)^{k}(\tfrac{1}{2}z)^% {k}}{k!}\mathop{\mathscr{C}_{{\nu\pm k}}\/}\nolimits\!\left(z\right),$ $|\lambda^{2}-1|<1$ .

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, : : factorial, : nonnegative integer, : 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
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If $\mathop{\mathscr{C}\/}\nolimits=\mathop{J\/}\nolimits$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary.

§10.23(ii) Addition Theorems

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 $\alpha=0,\pi$ .
Keywords:: Bessel functions, Hankel functions, cylinder functions
Referenced by:: §10.60(i), §28.27
Permalink:: http://dlmf.nist.gov/10.23.ii

¶ Neumann’s Addition Theorem

Keywords:: Bessel functions, Neumann’s addition theorem

10.23.2 $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(u\pm v\right)=\sum_{{k=-\infty}% }^{\infty}\mathop{\mathscr{C}_{{\nu\mp k}}\/}\nolimits\!\left(u\right)\mathop{% J_{{k}}\/}\nolimits\!\left(v\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, : 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
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The restriction is unnecessary when $\mathop{\mathscr{C}\/}\nolimits=\mathop{J\/}\nolimits$ and $\nu$ is an integer. Special cases are:

10.23.3 ${\mathop{J_{{0}}\/}\nolimits^{{2}}}\!\left(z\right)+2\sum_{{k=1}}^{\infty}{% \mathop{J_{{k}}\/}\nolimits^{{2}}}\!\left(z\right)=1,$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : nonnegative integer and : complex variable
A&S Ref:: 9.1.76
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.23.E3
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10.23.4 $\sum_{{k=0}}^{{2n}}(-1)^{k}\mathop{J_{{k}}\/}\nolimits\!\left(z\right)\mathop{% J_{{2n-k}}\/}\nolimits\!\left(z\right)\\ +2\sum_{{k=1}}^{\infty}\mathop{J_{{k}}\/}\nolimits\!\left(z\right)\mathop{J_{{% 2n+k}}\/}\nolimits\!\left(z\right)=0,$ $n\geq 1$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : integer, : nonnegative integer and : complex variable
A&S Ref:: 9.1.77
Permalink:: http://dlmf.nist.gov/10.23.E4
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10.23.5 $\sum_{{k=0}}^{n}\mathop{J_{{k}}\/}\nolimits\!\left(z\right)\mathop{J_{{n-k}}\/% }\nolimits\!\left(z\right)+2\sum_{{k=1}}^{\infty}(-1)^{k}\mathop{J_{{k}}\/}% \nolimits\!\left(z\right)\mathop{J_{{n+k}}\/}\nolimits\!\left(z\right)=\mathop% {J_{{n}}\/}\nolimits\!\left(2z\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : integer, : nonnegative integer and : complex variable
A&S Ref:: 9.1.78
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¶ Graf’s and Gegenbauer’s Addition Theorems

Keywords:: Bessel functions, Gegenbauer’s addition theorem, Graf’s addition theorem

Define

10.23.6

$w=\sqrt{u^{2}+v^{2}-2uv\mathop{\cos\/}\nolimits\alpha},$

$u-v\mathop{\cos\/}\nolimits\alpha=w\mathop{\cos\/}\nolimits\chi,$

$v\mathop{\sin\/}\nolimits\alpha=w\mathop{\sin\/}\nolimits\chi,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $\mathop{\sin\/}\nolimits z$ : sine function
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the branches being continuous and chosen so that $w\to u$ and $\chi\to 0$ as $v\to 0$ . If , are real and positive and $0\leq\alpha\leq\pi$ , then and $\chi$ are real and nonnegative, and the geometrical relationship is shown in Figure 10.23.1.

Figure 10.23.1: Graf’s and Gegenbauer’s addition theorems.

Referenced by:: ¶ ‣ §10.23(ii)
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10.23.7 $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(w\right)\selection{\mathop{\cos% \/}\nolimits\\ \mathop{\sin\/}\nolimits}(\nu\chi)=\sum_{{k=-\infty}}^{\infty}\mathop{\mathscr% {C}_{{\nu+k}}\/}\nolimits\!\left(u\right)\mathop{J_{{k}}\/}\nolimits\!\left(v% \right)\selection{\mathop{\cos\/}\nolimits\\ \mathop{\sin\/}\nolimits}(k\alpha),$ $|ve^{{\pm i\alpha}}|<|u|$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, : 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)
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10.23.8 $\frac{\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(w\right)}{w^{\nu}}=2^{\nu% }\mathop{\Gamma\/}\nolimits\!\left(\nu\right)\*\sum_{{k=0}}^{\infty}(\nu+k)% \frac{\mathop{\mathscr{C}_{{\nu+k}}\/}\nolimits\!\left(u\right)}{u^{\nu}}\frac% {\mathop{J_{{\nu+k}}\/}\nolimits\!\left(v\right)}{v^{\nu}}\mathop{C^{{(\nu)}}_% {{k}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right),$ $\nu\neq 0,-1,\dots$ , $|ve^{{\pm i\alpha}}|<|u|$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, : base of exponential function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, : 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)
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where $\mathop{C^{{(\nu)}}_{{k}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\alpha\right)$ is Gegenbauer’s polynomial (§18.3). The restriction $|ve^{{\pm i\alpha}}|<|u|$ is unnecessary in (10.23.7) when $\mathop{\mathscr{C}\/}\nolimits=\mathop{J\/}\nolimits$ and $\nu$ is an integer, and in (10.23.8) when $\mathop{\mathscr{C}\/}\nolimits=\mathop{J\/}\nolimits$ .

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

10.23.9 $e^{{iv\mathop{\cos\/}\nolimits\alpha}}=\frac{\mathop{\Gamma\/}\nolimits\!\left% (\nu\right)}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{{k=0}}^{\infty}(\nu+k)i^{k}\mathop{% J_{{\nu+k}}\/}\nolimits\!\left(v\right)\mathop{C^{{(\nu)}}_{{k}}\/}\nolimits\!% \left(\mathop{\cos\/}\nolimits\alpha\right),$ $\nu\neq 0,-1,\dots$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, : nonnegative integer and $\nu$ : complex parameter
A&S Ref:: 9.1.81
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¶ Partial Fractions

Keywords:: Bessel functions, Hankel functions

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

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).
Keywords:: Bessel functions
Referenced by:: §11.4(iv)
Permalink:: http://dlmf.nist.gov/10.23.iii

¶ Neumann’s Expansion

Keywords:: Bessel functions, Neumann’s expansion, Neumann’s polynomial

10.23.10 $f(z)=a_{0}\mathop{J_{{0}}\/}\nolimits\!\left(z\right)+2\sum_{{k=1}}^{\infty}a_% {k}\mathop{J_{{k}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : nonnegative integer, : complex variable, : function and $a_{k}$
A&S Ref:: 9.1.82
Referenced by:: §10.23(iii)
Permalink:: http://dlmf.nist.gov/10.23.E10
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where is the distance of the nearest singularity of the analytic function from ,

10.23.11 $a_{k}=\frac{1}{2\pi i}\int_{{|z|=c^{{\prime}}}}f(t)\mathop{O_{{k}}\/}\nolimits% \!\left(t\right)dt,$ $0<c^{{\prime}}<c$ ,

Symbols:: $\mathop{O_{{n}}\/}\nolimits\!\left(x\right)$ : Neumann’s polynomial, : differential of , $\int$ : integral, : nonnegative integer, : complex variable, : function and $a_{k}$
A&S Ref:: 9.1.83
Permalink:: http://dlmf.nist.gov/10.23.E11
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and $\mathop{O_{{k}}\/}\nolimits\!\left(t\right)$ is Neumann’s polynomial, defined by the generating function:

10.23.12 $\frac{1}{t-z}=\mathop{J_{{0}}\/}\nolimits\!\left(z\right)\mathop{O_{{0}}\/}% \nolimits\!\left(t\right)+2\sum_{{k=1}}^{\infty}\mathop{J_{{k}}\/}\nolimits\!% \left(z\right)\mathop{O_{{k}}\/}\nolimits\!\left(t\right),$

Defines:: $\mathop{O_{{n}}\/}\nolimits\!\left(x\right)$ : Neumann’s polynomial
Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : integer, : nonnegative integer, : real variable and : complex variable
A&S Ref:: 9.1.84
Permalink:: http://dlmf.nist.gov/10.23.E12
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$\mathop{O_{{n}}\/}\nolimits\!\left(t\right)$ is a polynomial of degree in $\ifrac{1}{t}:\mathop{O_{{0}}\/}\nolimits\!\left(t\right)=1/t$ and

10.23.13 $\mathop{O_{{n}}\/}\nolimits\!\left(t\right)=\frac{1}{4}\sum_{{k=0}}^{{\left% \lfloor n/2\right\rfloor}}\frac{(n-k-1)!n}{k!}\left(\frac{2}{t}\right)^{{n-2k+% 1}},$ $n=1,2,\ldots$ .

Symbols:: $\mathop{O_{{n}}\/}\nolimits\!\left(x\right)$ : Neumann’s polynomial, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , : integer and : nonnegative integer
A&S Ref:: 9.1.85
Referenced by:: §10.23(iii)
Permalink:: http://dlmf.nist.gov/10.23.E13
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For the more general form of expansion

10.23.14 $z^{\nu}f(z)=a_{0}\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)+2\sum_{{k=1}}^{% \infty}a_{k}\mathop{J_{{\nu+k}}\/}\nolimits\!\left(z\right)$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : nonnegative integer, : complex variable, $\nu$ : complex parameter, : function and $a_{k}$
A&S Ref:: 9.1.86
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see Watson (1944, §16.13), and for further generalizations see Watson (1944, Chapter 16) and Erdélyi et al. (1953b, §7.10.1).

¶ Examples

10.23.15 $(\tfrac{1}{2}z)^{\nu}=\sum_{{k=0}}^{\infty}\frac{(\nu+2k)\mathop{\Gamma\/}% \nolimits\!\left(\nu+k\right)}{k!}\mathop{J_{{\nu+2k}}\/}\nolimits\!\left(z% \right),$ $\nu\neq 0,-1,-2,\dots$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : : factorial, : nonnegative integer, : 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
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10.23.16 $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\left(\mathop{\ln\/}% \nolimits\!\left(\tfrac{1}{2}z\right)+\EulerConstant\right)\mathop{J_{{0}}\/}% \nolimits\!\left(z\right)-\frac{4}{\pi}\sum_{{k=1}}^{\infty}(-1)^{k}\frac{% \mathop{J_{{2k}}\/}\nolimits\!\left(z\right)}{k},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\EulerConstant$ : Euler’s constant, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer and : complex variable
A&S Ref:: 9.1.89
Permalink:: http://dlmf.nist.gov/10.23.E16
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10.23.17 $\mathop{Y_{{n}}\/}\nolimits\!\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{{-n}}}{% \pi}\sum_{{k=0}}^{{n-1}}\frac{(\tfrac{1}{2}z)^{k}\mathop{J_{{k}}\/}\nolimits\!% \left(z\right)}{k!(n-k)}+\frac{2}{\pi}\left(\mathop{\ln\/}\nolimits\!\left(% \tfrac{1}{2}z\right)-\mathop{\psi\/}\nolimits\!\left(n+1\right)\right)\mathop{% J_{{n}}\/}\nolimits\!\left(z\right)-\frac{2}{\pi}\sum_{{k=1}}^{\infty}(-1)^{k}% \frac{(n+2k)\mathop{J_{{n+2k}}\/}\nolimits\!\left(z\right)}{k(n+k)},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : integer, : nonnegative integer and : complex variable
A&S Ref:: 9.1.88
Referenced by:: §10.23(iii), §10.44(iii)
Permalink:: http://dlmf.nist.gov/10.23.E17
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where $\EulerConstant$ is Euler’s constant and $\mathop{\psi\/}\nolimits\!\left(n+1\right)={\mathop{\Gamma\/}\nolimits^{{% \prime}}}\!\left(n+1\right)/\mathop{\Gamma\/}\nolimits\!\left(n+1\right)$ (§5.2).

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

¶ Fourier–Bessel Expansion

Keywords:: Bessel functions, Fourier–Bessel expansion

Assume satisfies

10.23.18 $\int_{0}^{1}t^{{\frac{1}{2}}}|f(t)|dt<\infty,$

Symbols:: : differential of , $\int$ : integral and
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and define

10.23.19 $a_{m}=\frac{2}{(\mathop{J_{{\nu+1}}\/}\nolimits(\mathop{j_{{\nu,m}}\/}% \nolimits))^{2}}\int_{0}^{1}tf(t)\mathop{J_{{\nu}}\/}\nolimits\!\left(j_{{\nu,% m}}t\right)dt,$ $\nu\geq-\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : differential of , $\int$ : integral, $\mathop{j_{{\nu,m}}\/}\nolimits$ : zeros of the Bessel function $\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)$ , : integer, $\nu$ : complex parameter, $a_{k}$ and
Referenced by:: ¶ ‣ §10.74(vii)
Permalink:: http://dlmf.nist.gov/10.23.E19
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where $\mathop{j_{{\nu,m}}\/}\nolimits$ is as in §10.21(i). If , then

10.23.20 $\tfrac{1}{2}f(x-)+\tfrac{1}{2}f(x+)=\sum_{{m=1}}^{\infty}a_{m}\mathop{J_{{\nu}% }\/}\nolimits\!\left(j_{{\nu,m}}x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : integer, : real variable, $\nu$ : complex parameter, $a_{k}$ and
Permalink:: http://dlmf.nist.gov/10.23.E20
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provided that is of bounded variation (§1.4(v)) on an interval with . This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near and .

As an example,

10.23.21 $x^{\nu}=\sum_{{m=1}}^{\infty}\frac{2\!\mathop{J_{{\nu}}\/}\nolimits\!\left(j_{% {\nu,m}}x\right)}{\mathop{j_{{\nu,m}}\/}\nolimits\mathop{J_{{\nu+1}}\/}% \nolimits\!\left(j_{{\nu,m}}\right)},$ $\nu>0,0\leq x<1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{j_{{\nu,m}}\/}\nolimits$ : zeros of the Bessel function $\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)$ , : integer, : real variable and $\nu$ : complex parameter
Referenced by:: §10.23(iii)
Permalink:: http://dlmf.nist.gov/10.23.E21
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(Note that when the left-hand side is 1 and the right-hand side is 0.)

¶ Other Series Expansions

Keywords:: Bessel functions

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

Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.23.iv

For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§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).

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