10 Bessel Functions Bessel and Hankel Functions 10.21 Zeros 10.23 Sums

§10.22 Integrals

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Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.22
See also:: Annotations for Ch.10

§10.22(i) Indefinite Integrals
§10.22(ii) Integrals over Finite Intervals
§10.22(iii) Integrals over the Interval $(x,\infty)$
§10.22(iv) Integrals over the Interval $(0,\infty)$
§10.22(v) Hankel Transform
§10.22(vi) Compendia

§10.22(i) Indefinite Integrals

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Keywords:: cylinder functions, indefinite, integrals, integrals of Bessel and Hankel functions
Notes:: See Watson (1944, pp. 132–136). For (10.22.1)–(10.22.3) differentiate and use (10.6.2), (11.4.27), (11.4.28).
Permalink:: http://dlmf.nist.gov/10.22.i
See also:: Annotations for §10.22 and Ch.10

In this subsection $\mathscr{C}_{\nu}\left(z\right)$ and $\mathscr{D}_{\mu}(z)$ denote cylinder functions(§10.2(ii)) of orders $\nu$ and $\mu$ , respectively, not necessarily distinct.

10.22.1	$\displaystyle\int z^{\nu+1}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=z^{\nu+1}\mathscr{C}_{\nu+1}\left(z\right),$
10.22.1	$\displaystyle\int z^{-\nu+1}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=-z^{-\nu+1}\mathscr{C}_{\nu-1}\left(z\right).$
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 11.3.20, 11.3.21 (modified) Referenced by: §10.22(i) Permalink: http://dlmf.nist.gov/10.22.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.22(i), §10.22 and Ch.10

10.22.2		$\int z^{\nu}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z=\pi^{\frac{1}{2}}2^{% \nu-1}\Gamma\left(\nu+\tfrac{1}{2}\right)\*z\left(\mathscr{C}_{\nu}\left(z% \right)\mathbf{H}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu-1}\left(z\right)% \mathbf{H}_{\nu}\left(z\right)\right),$
$\nu\neq-\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 11.17 (Case $\nu=0$ only) Permalink: http://dlmf.nist.gov/10.22.E2 Encodings: TeX, pMML, png See also: Annotations for §10.22(i), §10.22 and Ch.10

For the Struve function $\mathbf{H}_{\nu}\left(z\right)$ see §11.2(i).

10.22.3		$\displaystyle\int e^{iz}z^{\nu}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=\frac{e^{iz}z^{\nu+1}}{2\nu+1}(\mathscr{C}_{\nu}\left(z\right)-i% \mathscr{C}_{\nu+1}\left(z\right)),$
	$\nu\neq-\tfrac{1}{2}$ ,
		$\displaystyle\int e^{iz}z^{-\nu}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z$	$\displaystyle=\frac{e^{iz}z^{-\nu+1}}{1-2\nu}(\mathscr{C}_{\nu}\left(z\right)+% i\mathscr{C}_{\nu-1}\left(z\right)),$
	$\nu\neq\tfrac{1}{2}$ .
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 11.3.9--11.3.11 Referenced by: §10.22(i) Permalink: http://dlmf.nist.gov/10.22.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.22(i), §10.22 and Ch.10

Products

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Keywords:: cylinder functions, indefinite, integrals, integrals of Bessel and Hankel functions, products
See also:: Annotations for §10.22(i), §10.22 and Ch.10

10.22.4		$\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(bz)\,\mathrm{d}z=\frac% {z\left(a\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\mu}(bz)-b\mathscr{C}_% {\mu}\left(az\right)\mathscr{D}_{\mu+1}(bz)\right)}{a^{2}-b^{2}},$
$a^{2}\neq b^{2}$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\mathscr{D}_{\nu}(z)$ : cylinder function Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E4 Encodings: TeX, pMML, png See also: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

10.22.5	$\displaystyle\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)\,% \mathrm{d}z$	$\displaystyle=\tfrac{1}{4}z^{2}\left(2\mathscr{C}_{\mu}\left(az\right)\mathscr% {D}_{\mu}(az)-\mathscr{C}_{\mu-1}\left(az\right)\mathscr{D}_{\mu+1}(az)-% \mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\mu-1}(az)\right),$
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\mathscr{D}_{\nu}(z)$ : cylinder function Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E5 Encodings: TeX, pMML, png See also: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10
10.22.6	$\displaystyle\int\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)\frac{\,% \mathrm{d}z}{z}$	$\displaystyle=-\frac{az(\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\nu}(az% )-\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu+1}(az))}{\mu^{2}-\nu^{2}}+% \frac{\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)}{\mu+\nu},$
$\mu^{2}\neq\nu^{2}$ ,
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function Permalink: http://dlmf.nist.gov/10.22.E6 Encodings: TeX, pMML, png See also: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

10.22.7		$\displaystyle\int z^{\mu+\nu+1}\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{% \nu}(az)\,\mathrm{d}z$	$\displaystyle=\frac{z^{\mu+\nu+2}}{2(\mu+\nu+1)}\*\left(\mathscr{C}_{\mu}\left% (az\right)\mathscr{D}_{\nu}(az)+\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_% {\nu+1}(az)\right),$
	$\mu+\nu\neq-1$ ,
		$\displaystyle\int z^{-\mu-\nu+1}\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{% \nu}(az)\,\mathrm{d}z$	$\displaystyle=\frac{z^{-\mu-\nu+2}}{2(1-\mu-\nu)}\*\left(\mathscr{C}_{\mu}% \left(az\right)\mathscr{D}_{\nu}(az)+\mathscr{C}_{\mu-1}\left(az\right)% \mathscr{D}_{\nu-1}(az)\right),$
	$\mu+\nu\neq 1$ .
ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function A&S Ref: 11.3.30, 11.3.31 Permalink: http://dlmf.nist.gov/10.22.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

§10.22(ii) Integrals over Finite Intervals

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Keywords:: integrals of Bessel and Hankel functions, over finite intervals
Notes:: For (10.22.8)–(10.22.12) see Luke (1962, pp. 51–53). To verify (10.22.13) construct the expansion of the left-hand side in powers of $z$ by use of (10.2.2), followed by term-by-term integration with the aid of (5.12.5) and (5.12.1). Then compare the result with the corresponding expansion of the right-hand side obtained from (10.8.3). Next, the result $\int_{0}^{2\pi}J_{2\nu}\left(2z\sin\theta\right)e^{\pm 2\mathrm{i}\mu\theta}\,% \mathrm{d}\theta=\pi e^{\pm\mathrm{i}\mu\pi}J_{\nu+\mu}\left(z\right)J_{\nu-% \mu}\left(z\right)$ , $\Re\nu>-\tfrac{1}{2}$ , is proved in a similar manner with the aid of (5.12.6) in place of (5.12.5)—from which (10.22.14) and (10.22.15) both follow. (10.22.17) follows by combining (10.22.13) and (10.2.3); (10.22.16) is a special case of (10.22.14). For (10.22.18) replace $\theta$ by $\tfrac{1}{2}\pi-\theta$ and set $\mu=n$ in (10.22.17); then apply (10.2.3) and let $\nu\to 0$ . For (10.22.19), (10.22.22), (10.22.25), (10.22.26) see Watson (1944, Chapter 12). (In the case of (10.22.25), page 374 of this reference lacks a factor $\frac{1}{2}$ on the right-hand side.) The verification of (10.22.20) is similar to that of (10.22.13), the role of (5.12.5) now being played by (5.12.2). For (10.22.21) combine (10.2.3) and (10.22.20). For (10.22.23) and (10.22.24) see Luke (1962, p. 302 (36) and p. 303 (39), respectively). For (10.22.27) see Watson (1944, p. 151). For (10.22.28), (10.22.29) differentiate and use (10.6.2). For (10.22.30) with $n\geq 1$ it follows by differentiation and use of (10.6.2) that the left-hand side equals $\int_{0}^{x}t^{-1}{J_{n}}^{2}\left(t\right)\,\mathrm{d}t-\tfrac{1}{2}{J_{n}}^{% 2}\left(x\right)$ ; application of Watson (1944, p. 152) yields the second result, then for the first result refer to (10.23.3). Some modifications of the proof of (10.22.30) are needed when $n=0$ . For (10.22.31)–(10.22.35) see Watson (1944, p. 380). For (10.22.36) replace $t$ by $z-t$ , substitute for $t^{\alpha}$ via (10.23.15) (with $z$ replaced by $t$ , and $\nu$ replaced by $\alpha$ ), and then apply (10.22.34). For (10.22.37) use (10.22.4) and (10.22.5); a similar proof applies to (10.22.38) after replacing $\mathscr{C}_{\mu\pm 1}\left(az\right)$ and $\mathscr{D}_{\mu\pm 1}(bz)$ by $\mp\mathscr{C}_{\mu}'\left(az\right)$ and $\mp\mathscr{D}_{\mu}^{\prime}(bz)$ , respectively, by means of (10.6.2).
Permalink:: http://dlmf.nist.gov/10.22.ii
See also:: Annotations for §10.22 and Ch.10

Throughout this subsection $x>0$ .

10.22.8		$\int_{0}^{x}J_{\nu}\left(t\right)\,\mathrm{d}t=2\sum_{k=0}^{\infty}J_{\nu+2k+1% }\left(x\right),$
$\Re\nu>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.1.2 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E8 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.9		$\int_{0}^{x}J_{2n}\left(t\right)\,\mathrm{d}t=\int_{0}^{x}J_{0}\left(t\right)% \,\mathrm{d}t-2\sum_{k=0}^{n-1}J_{2k+1}\left(x\right),\quad\int_{0}^{x}J_{2n+1% }\left(t\right)\,\mathrm{d}t=1-J_{0}\left(x\right)-2\sum_{k=1}^{n}J_{2k}\left(% x\right),$
$n=0,1,\dots$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer, $k$ : nonnegative integer and $x$ : real variable A&S Ref: 11.1.3, 11.1.4 Permalink: http://dlmf.nist.gov/10.22.E9 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.10		$\int_{0}^{x}t^{\mu}J_{\nu}\left(t\right)\,\mathrm{d}t=x^{\mu}\frac{\Gamma\left% (\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-% \frac{1}{2}\mu+\frac{1}{2}\right)}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\Gamma% \left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k\right)}{\Gamma\left(\frac{1}% {2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}J_{\nu+2k+1}\left(x\right),$
$\Re\left(\mu+\nu+1\right)>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.1.1 Permalink: http://dlmf.nist.gov/10.22.E10 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.11	$\displaystyle\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\,\mathrm{d}t$	$\displaystyle=\frac{1}{2}\sum_{k=1}^{\infty}\frac{\psi\left(k+1\right)-\psi% \left(1\right)}{k!}(\tfrac{1}{2}x)^{k}J_{k}\left(x\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\int$ : integral, $k$ : nonnegative integer and $x$ : real variable Referenced by: §10.22(iii), §10.43(ii) Permalink: http://dlmf.nist.gov/10.22.E11 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22 and Ch.10
10.22.12	$\displaystyle x\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\,\mathrm{d}t$	$\displaystyle=2\sum_{k=0}^{\infty}(2k+3)(\psi\left(k+2\right)-\psi\left(1% \right))J_{2k+3}\left(x\right)=x-2J_{1}\left(x\right)+2\sum_{k=0}^{\infty}(2k+% 5)\*(\psi\left(k+3\right)-\psi\left(1\right)-1)J_{2k+5}\left(x\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, $k$ : nonnegative integer and $x$ : real variable A&S Ref: 11.1.19 Referenced by: §10.22(ii), §10.22(iii), §10.43(ii) Permalink: http://dlmf.nist.gov/10.22.E12 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22 and Ch.10

where $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$ (§5.2(i)). See also (10.22.39).

Trigonometric Arguments

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Keywords:: integrals of Bessel and Hankel functions, trigonometric arguments
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.13	$\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{2\nu}\left(2z\cos\theta\right)\cos% \left(2\mu\theta\right)\,\mathrm{d}\theta$	$\displaystyle=\tfrac{1}{2}\pi J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z% \right),$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $n$ : integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 11.4.7 (Case $\nu=n$ , $\mu=0$ .) Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E13 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.14	$\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\cos\left(2\mu% \theta\right)\,\mathrm{d}\theta$	$\displaystyle=\pi\cos\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}% \left(z\right),$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $n$ : integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 11.4.8 (Case $\nu=0$ , $\mu=n$ .) Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E14 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.15	$\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\sin\left(2\mu% \theta\right)\,\mathrm{d}\theta$	$\displaystyle=\pi\sin\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}% \left(z\right),$
$\Re\nu>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E15 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.16	$\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{0}\left(2z\sin\theta\right)\cos\left(% 2n\theta\right)\,\mathrm{d}\theta$	$\displaystyle=\tfrac{1}{2}\pi{J_{n}}^{2}\left(z\right),$
$n=0,1,2,\dotsc$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E16 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.17		$\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos\theta\right)\cos\left(2\mu\theta% \right)\,\mathrm{d}\theta=\tfrac{1}{2}\pi\cot\left(2\nu\pi\right)J_{\nu+\mu}% \left(z\right)J_{\nu-\mu}\left(z\right)-\tfrac{1}{2}\pi\csc\left(2\nu\pi\right% )J_{\mu-\nu}\left(z\right)J_{-\mu-\nu}\left(z\right),$
$-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}$ ,
ⓘ 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, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E17 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.18		$\int_{0}^{\frac{1}{2}\pi}Y_{0}\left(2z\sin\theta\right)\cos\left(2n\theta% \right)\,\mathrm{d}\theta=\tfrac{1}{2}\pi J_{n}\left(z\right)Y_{n}\left(z% \right),$
$n=0,1,2,\dots$ .
ⓘ 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, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable A&S Ref: 11.4.9 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E18 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.19		$\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin\theta)^{\mu+1}(% \cos\theta)^{2\nu+1}\,\mathrm{d}\theta=2^{\nu}\Gamma\left(\nu+1\right)z^{-\nu-% 1}J_{\mu+\nu+1}\left(z\right),$
$\Re\mu>-1$ , $\Re\nu>-1$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 11.4.10 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E19 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.20	$\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin% \theta)^{\mu}(\cos\theta)^{2\mu}\,\mathrm{d}\theta$	$\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\Gamma\left(\mu+\tfrac{1}{2}% \right){J_{\mu}}^{2}\left(\tfrac{1}{2}z\right),$
$\Re\mu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $z$ : complex variable Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E20 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.21	$\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{\mu}\left(z\sin\theta\right)(\sin% \theta)^{\mu}(\cos\theta)^{2\mu}\,\mathrm{d}\theta$	$\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\Gamma\left(\mu+\tfrac{1}{2}% \right)J_{\mu}\left(\tfrac{1}{2}z\right)Y_{\mu}\left(\tfrac{1}{2}z\right),$
$\Re\mu>-\tfrac{1}{2}$ .
ⓘ 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, $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $z$ : complex variable Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E21 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.22		$\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin}^{2}\theta\right)J_{\nu}\left(z{% \cos}^{2}\theta\right)(\sin\theta)^{2\mu+1}(\cos\theta)^{2\nu+1}\,\mathrm{d}% \theta=\frac{\Gamma\left(\mu+\tfrac{1}{2}\right)\Gamma\left(\nu+\tfrac{1}{2}% \right)J_{\mu+\nu+\frac{1}{2}}\left(z\right)}{(8\pi z)^{\frac{1}{2}}\Gamma% \left(\mu+\nu+1\right)},$
$\Re\mu>-\tfrac{1}{2},\Re\nu>-\tfrac{1}{2}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E22 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.23	$\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin}^{2}\theta\right)J_{% \nu}\left(z{\cos}^{2}\theta\right)(\sin\theta)^{2\alpha-1}\sec\theta\,\mathrm{% d}\theta$	$\displaystyle=\frac{(\mu+\nu+\alpha)\Gamma\left(\mu+\alpha\right)2^{\alpha-1}}% {\nu\Gamma\left(\mu+1\right)z^{\alpha}}J_{\mu+\nu+\alpha}\left(z\right),$
$\Re\left(\mu+\alpha\right)>0$ , $\Re\nu>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma 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, $\Re$ : real part, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $\nu$ : complex parameter and $\alpha$ : positive integer A&S Ref: 11.4.11 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E23 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.24	$\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin}^{2}\theta\right)J_{% \nu}\left(z{\cos}^{2}\theta\right)\cot\theta\,\mathrm{d}\theta$	$\displaystyle=\tfrac{1}{2}\mu^{-1}J_{\mu+\nu}\left(z\right),$
$\Re\mu>0,\Re\nu>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E24 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.25	$\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)I_{\nu}% \left(z\cos\theta\right)(\tan\theta)^{\mu+1}\,\mathrm{d}\theta$	$\displaystyle=\frac{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{% 1}{2}z)^{\mu}}{2\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}J_{\nu}% \left(z\right),$
$\Re\nu>\Re\mu>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma 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, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E25 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

For $I_{\nu}$ see §10.25(ii).

10.22.26		$\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)J_{\nu}\left(\zeta\cos% \theta\right)(\sin\theta)^{\mu+1}(\cos\theta)^{\nu+1}\,\mathrm{d}\theta=\frac{% z^{\mu}\zeta^{\nu}J_{\mu+\nu+1}\left(\sqrt{\zeta^{2}+z^{2}}\right)}{(\zeta^{2}% +z^{2})^{\frac{1}{2}(\mu+\nu+1)}},$
$\Re\mu>-1,\Re\nu>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E26 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

Products

ⓘ

See also:: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.27	$\displaystyle\int_{0}^{x}t{J_{\nu-1}}^{2}\left(t\right)\,\mathrm{d}t$	$\displaystyle=2\sum_{k=0}^{\infty}(\nu+2k){J_{\nu+2k}}^{2}\left(x\right),$
$\Re\nu>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.3.32 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E27 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.28	$\displaystyle\int_{0}^{x}t\left({J_{\nu-1}}^{2}\left(t\right)-{J_{\nu+1}}^{2}% \left(t\right)\right)\,\mathrm{d}t$	$\displaystyle=2\nu{J_{\nu}}^{2}\left(x\right),$
$\Re\nu>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.3.33 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E28 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.29	$\displaystyle\int_{0}^{x}t{J_{0}}^{2}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\tfrac{1}{2}x^{2}\left({J_{0}}^{2}\left(x\right)+{J_{1}}^{2}% \left(x\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $x$ : real variable A&S Ref: 11.3.34 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E29 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.30		$\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}% \left(1-{J_{0}}^{2}\left(x\right)\right)-\sum_{k=1}^{n}{J_{k}}^{2}\left(x% \right)=\sum_{k=n+1}^{\infty}{J_{k}}^{2}\left(x\right),$
$n=0,1,2,\dotsc$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer, $k$ : nonnegative integer and $x$ : real variable A&S Ref: 11.3.35 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E30 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

Convolutions

ⓘ

Keywords:: convolutions, integrals of Bessel and Hankel functions
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.31		$\int_{0}^{x}J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\,\mathrm{d}t=2\sum_{k% =0}^{\infty}(-1)^{k}J_{\mu+\nu+2k+1}\left(x\right),$
$\Re\mu>-1,\Re\nu>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.3.37 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E31 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.32	$\displaystyle\int_{0}^{x}J_{\nu}\left(t\right)J_{1-\nu}\left(x-t\right)\,% \mathrm{d}t$	$\displaystyle=J_{0}\left(x\right)-\cos x,$
$-1<\Re\nu<2$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.3.38 Permalink: http://dlmf.nist.gov/10.22.E32 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10
10.22.33	$\displaystyle\int_{0}^{x}J_{\nu}\left(t\right)J_{-\nu}\left(x-t\right)\,% \mathrm{d}t$	$\displaystyle=\sin x,$
$\|\Re\nu\|<1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.3.39 Permalink: http://dlmf.nist.gov/10.22.E33 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.34		$\int_{0}^{x}t^{-1}J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\,\mathrm{d}t=% \frac{J_{\mu+\nu}\left(x\right)}{\mu},$
$\Re\mu>0,\Re\nu>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.3.40 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E34 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.35		$\int_{0}^{x}\frac{J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\,\mathrm{d}t}{t% (x-t)}=\frac{(\mu+\nu)J_{\mu+\nu}\left(x\right)}{\mu\nu x},$
$\Re\mu>0,\Re\nu>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 11.3.41 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E35 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

Fractional Integral

ⓘ

Keywords:: fractional, integrals of Bessel and Hankel functions
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.36		$\frac{1}{\Gamma\left(\alpha\right)}\int_{0}^{x}(x-t)^{\alpha-1}J_{\nu}\left(t% \right)\,\mathrm{d}t=2^{\alpha}\sum_{k=0}^{\infty}\frac{{\left(\alpha\right)_{% k}}}{k!}J_{\nu+\alpha+2k}\left(x\right),$
$\Re\alpha>0,\Re\nu\geq 0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $x$ : real variable, $\nu$ : complex parameter and $\alpha$ : positive integer A&S Ref: 11.2.3 and 11.2.4 Referenced by: §10.22(ii), §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E36 Encodings: TeX, pMML, png Correction (effective with 1.1.2): The Pochhammer symbol now links to its definition. See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

When $\alpha=m=1,2,3,\ldots$ the left-hand side of (10.22.36) is the $m$ th repeated integral of $J_{\nu}\left(x\right)$ (§§1.4(v) and 1.15(vi)).

Orthogonality

ⓘ

Keywords:: Bessel functions, integrals of Bessel and Hankel functions, orthogonal properties, orthogonality, over finite intervals
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

If $\nu>-1$ , then

10.22.37		$\int_{0}^{1}tJ_{\nu}\left(j_{\nu,\ell}t\right)J_{\nu}\left(j_{\nu,m}t\right)\,% \mathrm{d}t=\tfrac{1}{2}\left(J_{\nu}'\left(j_{\nu,\ell}\right)\right)^{2}% \delta_{\ell,m},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer and $\nu$ : complex parameter A&S Ref: 11.4.5 Referenced by: §10.22(ii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9) Permalink: http://dlmf.nist.gov/10.22.E37 Encodings: TeX, pMML, png Clarification (effective with 1.0.17): The Kronecker delta symbol has been moved furthest to the right. See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

where $j_{\nu,\ell}$ and $j_{\nu,m}$ are zeros of $J_{\nu}\left(x\right)$ (§10.21(i)), and $\delta_{\ell,m}$ is Kronecker’s symbol.

Also, if $a,b,\nu$ are real constants with $b\neq 0$ and $\nu>-1$ , then

10.22.38		$\int_{0}^{1}tJ_{\nu}\left(\alpha_{\ell}t\right)J_{\nu}\left(\alpha_{m}t\right)% \,\mathrm{d}t=\left(\frac{a^{2}}{b^{2}}+\alpha_{\ell}^{2}-\nu^{2}\right)\frac{% (J_{\nu}\left(\alpha_{\ell}\right))^{2}}{2\alpha_{\ell}^{2}}\delta_{\ell,m},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer and $\nu$ : complex parameter A&S Ref: 11.4.5 Referenced by: §10.22(ii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9) Permalink: http://dlmf.nist.gov/10.22.E38 Encodings: TeX, pMML, png Clarification (effective with 1.0.17): The Kronecker delta symbol has been moved furthest to the right. See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

where $\alpha_{\ell}$ and $\alpha_{m}$ are positive zeros of $aJ_{\nu}\left(x\right)+bxJ_{\nu}'\left(x\right)$ . (Compare (10.22.55)).

§10.22(iii) Integrals over the Interval $(x,\infty)$

ⓘ

Keywords:: integrals of Bessel and Hankel functions, over infinite intervals
Notes:: For the first result in (10.22.39) use (10.22.43) with $\nu=0$ and $\mu$ replaced by $\mu-1$ , split the integration range at $t=x$ and take limits as $\mu\to 0$ ; for the second result substitute into the first result by (10.2.2) and integrate term by term. (10.22.40), is proved in a similar manner, starting from (10.22.44) and substituting by means of (10.8.2) and (10.2.2) with $\nu=0$ for the term-by-term integration.
Permalink:: http://dlmf.nist.gov/10.22.iii
See also:: Annotations for §10.22 and Ch.10

When $x>0$

10.22.39		$\int_{x}^{\infty}\frac{J_{0}\left(t\right)}{t}\,\mathrm{d}t+\gamma+\ln\left(% \tfrac{1}{2}x\right)=\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\,\mathrm{d}t=% \sum_{k=1}^{\infty}(-1)^{k-1}\frac{(\frac{1}{2}x)^{2k}}{2k(k!)^{2}},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\gamma$ : Euler’s constant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $x$ : real variable A&S Ref: 11.1.20 Referenced by: §10.22(ii), §10.22(iii), §10.43(ii) Permalink: http://dlmf.nist.gov/10.22.E39 Encodings: TeX, pMML, png See also: Annotations for §10.22(iii), §10.22 and Ch.10

10.22.40		$\int_{x}^{\infty}\frac{Y_{0}\left(t\right)}{t}\,\mathrm{d}t=-\frac{1}{\pi}% \left(\ln\left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+\frac{\pi}{6}+\frac{2}{% \pi}\sum_{k=1}^{\infty}(-1)^{k}\*\left(\psi\left(k+1\right)+\frac{1}{2k}-\ln% \left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}},$
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $x$ : real variable A&S Ref: 11.1.21 Referenced by: §10.22(iii), §10.43(ii) Permalink: http://dlmf.nist.gov/10.22.E40 Encodings: TeX, pMML, png See also: Annotations for §10.22(iii), §10.22 and Ch.10

where $\gamma$ is Euler’s constant (§5.2(ii)). Compare (10.22.11) and (10.22.12).

§10.22(iv) Integrals over the Interval $(0,\infty)$

ⓘ

Notes:: For (10.22.41)–(10.22.45) see Luke (1962, pp. 56–57). For (10.22.46) see Erdélyi et al. (1953b, p. 96). (10.22.47) is the special case of Eq. (6) of Watson (1944, §13.53) obtained by setting $\mu=b=0$ , $\rho=\nu+1$ , and subsequently replacing $k$ by $b$ . For (10.22.48) see Sneddon (1966, Eq. (2.1.32)). For (10.22.49)–(10.22.59) see Watson (1944, pp. 385, 394, 403–405, 407; there is an error in Eq. (1), p. 407). For (10.22.60) differentiate (10.22.59) with respect to $\mu$ and use (10.2.4) with $n=0$ . For (10.22.61) see Watson (1944, p. 405). (10.22.62) follows from (10.22.56) with $\lambda=\nu-\mu-1$ and (15.4.6). For (10.22.63), (10.22.64) see Watson (1944, p. 404). For (10.22.65) apply (10.22.56) with $\mu=\nu=0$ , then let $\lambda\to 1$ . For (10.22.66), (10.22.67) see Watson (1944, pp. 389, 395). For (10.22.68) set $a=b$ in (10.22.67), differentiate with respect to $\nu$ and apply (10.2.4) and (10.27.5) with $n=0$ . For (10.22.69), (10.22.70), see Watson (1944, p. 429, Eqs. (3),(4), with $\mu=\nu+1$ in (3)). For (10.22.71), (10.22.72) see Watson (1944, pp. 411, 412). For (10.22.74), (10.22.75) see Watson (1944, p. 411) and Askey et al. (1986).
Permalink:: http://dlmf.nist.gov/10.22.iv
See also:: Annotations for §10.22 and Ch.10

10.22.41	$\displaystyle\int_{0}^{\infty}J_{\nu}\left(t\right)\,\mathrm{d}t$	$\displaystyle=1,$
$\Re\nu>-1$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter A&S Ref: 11.4.17 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E41 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10
10.22.42	$\displaystyle\int_{0}^{\infty}Y_{\nu}\left(t\right)\,\mathrm{d}t$	$\displaystyle=-\tan\left(\tfrac{1}{2}\nu\pi\right),$
$\|\Re\nu\|<1$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\tan\NVar{z}$ : tangent function and $\nu$ : complex parameter A&S Ref: 11.4.20, 11.4.21 Permalink: http://dlmf.nist.gov/10.22.E42 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.43	$\displaystyle\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)\,\mathrm{d}t$	$\displaystyle=2^{\mu}\frac{\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{% 1}{2}\right)}{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},$
$\Re\left(\mu+\nu\right)>-1$ , $\Re\mu<\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter Keywords: Mellin transform A&S Ref: 11.4.16 Referenced by: §10.22(iii), §10.59 Permalink: http://dlmf.nist.gov/10.22.E43 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10
10.22.44	$\displaystyle\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\frac{2^{\mu}}{\pi}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+% \tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}% \right)\sin\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi,$
$\Re\left(\mu\pm\nu\right)>-1$ , $\Re\mu<\tfrac{1}{2}$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $\nu$ : complex parameter Keywords: Mellin transform A&S Ref: 11.4.19 (modified) Referenced by: §10.22(iii) Permalink: http://dlmf.nist.gov/10.22.E44 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.45		$\int_{0}^{\infty}\frac{1-J_{0}\left(t\right)}{t^{\mu}}\,\mathrm{d}t=-\frac{\pi% \sec\left(\frac{1}{2}\mu\pi\right)}{2^{\mu}{\Gamma}^{2}\left(\frac{1}{2}\mu+% \frac{1}{2}\right)},$
$1<\Re\mu<3$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\sec\NVar{z}$ : secant function Keywords: Mellin transform A&S Ref: 11.4.18 (modified) Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E45 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.46		$\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at\right)}{(t^{2}+b^{2})^{\mu+1}}% \,\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\Gamma\left(\mu+1\right)}K_{\nu% -\mu}\left(ab\right),$
$a>0$ , $\Re b>0$ , $-1<\Re\nu<2\Re\mu+\tfrac{3}{2}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and $\nu$ : complex parameter Keywords: Mellin transform A&S Ref: 11.4.44 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E46 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.47		$\int_{0}^{\infty}\frac{t^{\nu}Y_{\nu}\left(at\right)}{t^{2}+b^{2}}\,\mathrm{d}% t=-b^{\nu-1}K_{\nu}\left(ab\right),$
$a>0,\Re b>0,-\tfrac{1}{2}<\Re\nu<\tfrac{5}{2}$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and $\nu$ : complex parameter Keywords: Mellin transform A&S Ref: 11.4.46 (is the special case $\nu=0$ .) Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E47 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

For $K_{\nu}$ see §10.25(ii).

10.22.48		$\int_{0}^{\infty}J_{\mu}\left(x\cosh\phi\right)(\cosh\phi)^{1-\mu}(\sinh\phi)^% {2\nu+1}\,\mathrm{d}\phi=2^{\nu}\Gamma\left(\nu+1\right)x^{-\nu-1}J_{\mu-\nu-1% }\left(x\right),$
$x>0,\Re\nu>-1,\Re\mu>2\Re\nu+\tfrac{1}{2}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, $x$ : real variable and $\nu$ : complex parameter Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E48 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.49		$\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt\right)\,\mathrm{d}t=\frac{(% \tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+\nu\right)\*\mathbf{F}\left(% \frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),$
$\Re\left(\mu+\nu\right)>0,\Re\left(a\pm ib\right)>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter Keywords: Laplace transform, Mellin transform Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E49 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.50		$\int_{0}^{\infty}t^{\mu-1}e^{-at}Y_{\nu}\left(bt\right)\,\mathrm{d}t=\cot\left% (\nu\pi\right)\frac{(\tfrac{1}{2}b)^{\nu}\Gamma\left(\mu+\nu\right)}{(a^{2}+b^% {2})^{\frac{1}{2}(\mu+\nu)}}\\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{1-\mu+% \nu}{2};\nu+1;\frac{b^{2}}{a^{2}+b^{2}}\right)-\csc\left(\nu\pi\right)\frac{(% \tfrac{1}{2}b)^{-\nu}\Gamma\left(\mu-\nu\right)}{(a^{2}+b^{2})^{\frac{1}{2}(% \mu-\nu)}}\\mathbf{F}\left(\frac{\mu-\nu}{2},\frac{1-\mu-\nu}{2};1-\nu;\frac{% b^{2}}{a^{2}+b^{2}}\right),$
$\Re\mu>\|\Re\nu\|,\Re\left(a\pm ib\right)>0$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.22.E50 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

For the hypergeometric function $\mathbf{F}$ see §15.2(i).

10.22.51	$\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}% \right)t^{\nu+1}\,\mathrm{d}t$	$\displaystyle=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp\left(-\frac{b^{2}}{4p^{2}}% \right),$
$\Re\nu>-1$ , $\Re\left(p^{2}\right)>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter A&S Ref: 11.4.29 Permalink: http://dlmf.nist.gov/10.22.E51 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10
10.22.52	$\displaystyle\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}% \right)\,\mathrm{d}t$	$\displaystyle=\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p^{2}}\right)I_{% \ifrac{\nu}{2}}\left(\frac{b^{2}}{8p^{2}}\right),$
$\Re\nu>-1,\Re\left(p^{2}\right)>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.22.E52 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.53		$\int_{0}^{\infty}Y_{2\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)\,\mathrm{% d}t=-\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p^{2}}\right)\left(I_{\nu}% \left(\frac{b^{2}}{8p^{2}}\right)\tan\left(\nu\pi\right)+\frac{1}{\pi}K_{\nu}% \left(\frac{b^{2}}{8p^{2}}\right)\sec\left(\nu\pi\right)\right),$
$\|\Re\nu\|<\tfrac{1}{2}$ , $\Re\left(p^{2}\right)>0$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, $\sec\NVar{z}$ : secant function, $\tan\NVar{z}$ : tangent function and $\nu$ : complex parameter A&S Ref: 11.4.30 Permalink: http://dlmf.nist.gov/10.22.E53 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

For $I$ and $K$ see §10.25(ii).

10.22.54		$\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)t^{\mu-1}\,% \mathrm{d}t=\frac{(\tfrac{1}{2}b/p)^{\nu}\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}% {2}\mu\right)}{2p^{\mu}}\exp\left(-\frac{b^{2}}{4p^{2}}\right)\*{\mathbf{M}}% \left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^{2}}{4p^{2}}\right),$
$\Re\left(\mu+\nu\right)>0$ , $\Re\left(p^{2}\right)>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter A&S Ref: 11.4.28 (modified) Permalink: http://dlmf.nist.gov/10.22.E54 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

For the confluent hypergeometric function ${\mathbf{M}}$ see §13.2(i).

Orthogonality

ⓘ

Keywords:: Bessel functions, integrals of Bessel and Hankel functions, orthogonal properties, orthogonality
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.55		$\int_{0}^{\infty}t^{-1}J_{\nu+2\ell+1}\left(t\right)J_{\nu+2m+1}\left(t\right)% \,\mathrm{d}t=\frac{\delta_{\ell,m}}{2(2\ell+\nu+1)},$
$\nu+\ell+m>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer and $\nu$ : complex parameter A&S Ref: 11.4.6 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E55 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

ⓘ

Keywords:: Bessel functions, Weber–Schafheitlin discontinuous integrals
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.56		$\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{\nu}\left(bt\right)}{t^{% \lambda}}\,\mathrm{d}t=\frac{a^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu-% \frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{\mu-\lambda+1}\Gamma\left% (\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{1}{2}\right)}\*\mathbf% {F}\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(\mu-\nu-\lambda+1);\mu+1% ;\frac{a^{2}}{b^{2}}\right),$
$0<a<b$ , $\Re\left(\mu+\nu+1\right)>\Re\lambda>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter Keywords: Mellin transform A&S Ref: 11.4.33, 11.4.34 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E56 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

If $0<b<a$ , then interchange $a$ and $b$ , and also $\mu$ and $\nu$ . If $b=a$ , then

10.22.57	$\displaystyle\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{\nu}\left(at% \right)}{t^{\lambda}}\,\mathrm{d}t$	$\displaystyle=\frac{(\frac{1}{2}a)^{\lambda-1}\Gamma\left(\frac{1}{2}\mu+\frac% {1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\right)\Gamma\left(\lambda\right)}{2% \Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right% )\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}% \right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\nu+\frac{1}{2% }\right)},$
$\Re\left(\mu+\nu+1\right)>\Re\lambda>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.22.E57 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10
10.22.58	$\displaystyle\int_{0}^{\infty}\frac{J_{\nu}\left(at\right)J_{\nu}\left(bt% \right)}{t^{\lambda}}\,\mathrm{d}t$	$\displaystyle=\frac{(ab)^{\nu}\Gamma\left(\nu-\frac{1}{2}\lambda+\frac{1}{2}% \right)}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{2}\lambda+\frac{1}{2}}\Gamma% \left(\frac{1}{2}\lambda+\frac{1}{2}\right)}\mathbf{F}\left(\frac{2\nu+1-% \lambda}{4},\frac{2\nu+3-\lambda}{4};\nu+1;\frac{4a^{2}b^{2}}{(a^{2}+b^{2})^{2% }}\right),$
$a\neq b$ , $\Re\left(2\nu+1\right)>\Re\lambda>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.22.E58 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\mu>-1$

10.22.59		$\int_{0}^{\infty}e^{ibt}J_{\mu}\left(at\right)\,\mathrm{d}t=\begin{cases}% \dfrac{\exp\left(i\mu\operatorname{arcsin}\left(b/a\right)\right)}{(a^{2}-b^{2% })^{\frac{1}{2}}},&0\leq b<a,\\ \dfrac{ia^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{(b^{2}-a^{2})^{\frac{1}{2% }}\left(b+(b^{2}-a^{2})^{\frac{1}{2}}\right)^{\mu}},&0<a<b.\end{cases}$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{arcsin}\NVar{z}$ : arcsine function A&S Ref: 11.4.37, 11.4.38 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E59 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

10.22.60		$\int_{0}^{\infty}e^{ibt}Y_{0}\left(at\right)\,\mathrm{d}t=\begin{cases}(2i/\pi% )(a^{2}-b^{2})^{-\frac{1}{2}}\operatorname{arcsin}\left(b/a\right),&0\leq b<a,% \\ (b^{2}-a^{2})^{-\frac{1}{2}}\left(-1+\dfrac{2i}{\pi}\ln\left(\dfrac{a}{b+(b^{2% }-a^{2})^{\frac{1}{2}}}\right)\right),&0<a<b.\end{cases}$
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 11.4.40 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E60 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\mu>0$ ,

10.22.61		$\int_{0}^{\infty}t^{-1}e^{ibt}J_{\mu}\left(at\right)\,\mathrm{d}t=\begin{cases% }(1/\mu)\exp\left(i\mu\operatorname{arcsin}\left(b/a\right)\right),&0\leq b% \leq a,\\ \dfrac{a^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{\mu\left(b+(b^{2}-a^{2})^{% \frac{1}{2}}\right)^{\mu}},&0<a\leq b.\end{cases}$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{arcsin}\NVar{z}$ : arcsine function Keywords: Mellin transform A&S Ref: 11.4.35, 11.4.36 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E61 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\nu>\Re\mu>-1$ ,

10.22.62		$\int_{0}^{\infty}t^{\mu-\nu+1}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)\,% \mathrm{d}t=\begin{cases}0,&0<b<a,\\ \dfrac{2^{\mu-\nu+1}a^{\mu}(b^{2}-a^{2})^{\nu-\mu-1}}{b^{\nu}\Gamma\left(\nu-% \mu\right)},&0<a\leq b.\end{cases}$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\nu$ : complex parameter Keywords: Mellin transform A&S Ref: 11.4.41 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E62 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\mu>0$ ,

10.22.63		$\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\mu-1}\left(bt\right)\,\mathrm{d}t=% \begin{cases}b^{\mu-1}a^{-\mu},&0<b<a,\\ (2b)^{-1},&b=a(>0),\\ 0,&0<a<b.\end{cases}$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral A&S Ref: 11.4.42 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E63 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $n=0,1,2,\dots$ and $\Re\mu>-n-1$ ,

10.22.64		$\int_{0}^{\infty}J_{\mu+2n+1}\left(at\right)J_{\mu}\left(bt\right)\,\mathrm{d}% t=\begin{cases}\dfrac{b^{\mu}\Gamma\left(\mu+n+1\right)}{a^{\mu+1}n!}\mathbf{F% }\left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}\right),&0<b<a,\\ (-1)^{n}/(2a),&b=a(>0),\\ 0,&0<a<b.\end{cases}$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $n$ : integer Referenced by: §10.22(iv), §10.59 Permalink: http://dlmf.nist.gov/10.22.E64 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

10.22.65		$\int_{0}^{\infty}J_{0}\left(at\right)\left(J_{0}\left(bt\right)-J_{0}\left(ct% \right)\right)\frac{\,\mathrm{d}t}{t}=\begin{cases}0,&0\leq b<a,0<c\leq a,\\ \ln\left(c/a\right),&0\leq b<a\leq c.\end{cases}$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 11.4.43 (Case $b=0$ .) Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E65 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

Other Double Products

ⓘ

See also:: Annotations for §10.22(iv), §10.22 and Ch.10

In (10.22.66)–(10.22.70) $a, b, c$ are positive constants.

10.22.66	$\displaystyle\int_{0}^{\infty}e^{-at}J_{\nu}\left(bt\right)J_{\nu}\left(ct% \right)\,\mathrm{d}t$	$\displaystyle=\frac{1}{\pi(bc)^{\frac{1}{2}}}\*Q_{\nu-\frac{1}{2}}\left(\frac{% a^{2}+b^{2}+c^{2}}{2bc}\right),$
$\Re\nu>-\tfrac{1}{2}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind and $\nu$ : complex parameter Keywords: Laplace transform Referenced by: §10.22(iv), §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E66 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10
10.22.67	$\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{\nu}\left(at% \right)J_{\nu}\left(bt\right)\,\mathrm{d}t$	$\displaystyle=\frac{1}{2p^{2}}\exp\left(-\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{% \nu}\left(\frac{ab}{2p^{2}}\right),$
$\Re\nu>-1,\Re\left(p^{2}\right)>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part and $\nu$ : complex parameter Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E67 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10
10.22.68	$\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{0}\left(at\right% )Y_{0}\left(at\right)\,\mathrm{d}t$	$\displaystyle=-\frac{1}{2\pi p^{2}}\exp\left(-\frac{a^{2}}{2p^{2}}\right)K_{0}% \left(\frac{a^{2}}{2p^{2}}\right),$
$\Re\left(p^{2}\right)>0$ .
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $\Re$ : real part Referenced by: §10.22(iv), §10.43(iv) Permalink: http://dlmf.nist.gov/10.22.E68 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

For the associated Legendre function $Q$ see §14.3(ii) with $\mu=0$ . For $I$ and $K$ see §10.25(ii).

10.22.69	$\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}\left(bt\right)% \frac{t\,\mathrm{d}t}{t^{2}-z^{2}}$	$\displaystyle=\left\{\begin{array}[]{ll}\frac{1}{2}\pi iJ_{\nu}\left(bz\right)% {H^{(1)}_{\nu}}\left(az\right),&a>b\\ \frac{1}{2}\pi iJ_{\nu}\left(az\right){H^{(1)}_{\nu}}\left(bz\right),&b>a\end{% array}\right\},$
$\Re\nu>-1,\Im z>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E69 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10
10.22.70	$\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu+1}\left(bt\right)% \frac{t\,\mathrm{d}t}{t^{2}-z^{2}}$	$\displaystyle=\frac{1}{2}\pi J_{\nu+1}\left(bz\right){H^{(1)}_{\nu}}\left(az% \right),$
$a\geq b>0$ , $\Re\nu>-\tfrac{3}{2},\Im z>0$ .
ⓘ 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, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Im$ : imaginary part, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(iv), §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E70 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

Equation (10.22.70) also remains valid if the order $\nu+1$ of the $J$ functions on both sides is replaced by $\nu+2n-3$ , $n=1,2,\dots$ , and the constraint $\Re\nu>-\frac{3}{2}$ is replaced by $\Re\nu>-n+\frac{1}{2}$ .

See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions.

Triple Products

ⓘ

Keywords:: area of triangle, integrals of Bessel and Hankel functions, over infinite intervals, products, triple
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

In (10.22.71) and (10.22.72) $a, b, c$ are positive constants.

10.22.71	$\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)t^{1-\mu}\,\mathrm{d}t$	$\displaystyle=\frac{(bc)^{\mu-1}(\sin\phi)^{\mu-\frac{1}{2}}}{(2\pi)^{\frac{1}% {2}}a^{\mu}}\mathsf{P}^{\frac{1}{2}-\mu}_{\nu-\frac{1}{2}}\left(\cos\phi\right),$
$\Re\mu>-\tfrac{1}{2},\Re\nu>-1,\|b-c\|<a<b+c,\cos\phi=(b^{2}+c^{2}-a^{2})/(2bc)$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : 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, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $\nu$ : complex parameter Keywords: Mellin transform Referenced by: §10.22(iv), §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E71 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10
10.22.72	$\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)t^{1-\mu}\,\mathrm{d}t$	$\displaystyle=\frac{(bc)^{\mu-1}\sin\left((\mu-\nu)\pi\right)(\sinh\chi)^{\mu-% \frac{1}{2}}}{(\frac{1}{2}\pi^{3})^{\frac{1}{2}}a^{\mu}}{\mathrm{e}}^{(\mu-% \frac{1}{2})\mathrm{i}\pi}Q^{\frac{1}{2}-\mu}_{\nu-\frac{1}{2}}\left(\cosh\chi% \right),$
$\Re\mu>-\tfrac{1}{2},\Re\nu>-1,a>b+c,\cosh\chi=(a^{2}-b^{2}-c^{2})/(2bc)$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $\nu$ : complex parameter Keywords: Mellin transform Referenced by: §10.22(iv), §10.22(iv), Erratum (V1.0.21) for Equation (10.22.72) Permalink: http://dlmf.nist.gov/10.22.E72 Encodings: TeX, pMML, png Clarification (effective with 1.0.21): Originally, the factor on the right-hand side was written as $\frac{(bc)^{\mu-1}\cos\left(\nu\pi\right)(\sinh\chi)^{\mu-\frac{1}{2}}}{(\frac% {1}{2}\pi^{3})^{\frac{1}{2}}a^{\mu}}$ , which was taken directly from Watson (1944, p. 412, (13.46.5)), who uses a different normalization for the associated Legendre functions of the second kind $Q^{\mu}_{\nu}$ . Watson’s $Q_{\nu}^{\mu}$ equals $\frac{\sin\left((\nu+\mu)\pi\right)}{\sin\left(\nu\pi\right)}{\mathrm{e}}^{-% \mu\pi\mathrm{i}}Q^{\mu}_{\nu}$ in the DLMF. See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

For the Ferrers function $\mathsf{P}$ and the associated Legendre function $Q$ , see §§14.3(i) and 14.3(ii), respectively.

In (10.22.74) and (10.22.75), $a, b, c$ are positive constants and

10.22.73	$\displaystyle A$	$\displaystyle=s(s-a)(s-b)(s-c),$
10.22.73	$\displaystyle s$	$\displaystyle=\tfrac{1}{2}(a+b+c).$
ⓘ Defines: $A$ : area (locally) and $s$ : sum (locally) Permalink: http://dlmf.nist.gov/10.22.E73 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

(Thus if $a, b, c$ are the sides of a triangle, then $A^{\frac{1}{2}}$ is the area of the triangle.)

If $\Re\nu>-\tfrac{1}{2}$ , then

10.22.74	$\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)t^{1-\nu}\,\mathrm{d}t$	$\displaystyle=\begin{cases}\dfrac{2^{\nu-1}A^{\nu-\frac{1}{2}}}{\pi^{\frac{1}{% 2}}(abc)^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)},&A>0,\\ 0,&A\leq 0.\end{cases}$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\nu$ : complex parameter and $A$ : area Keywords: Mellin transform Referenced by: §10.22(iv), §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E74 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10
If $\|\nu\|<\tfrac{1}{2}$ , then
10.22.75	$\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)t^{1+\nu}\,\mathrm{d}t$	$\displaystyle=\begin{cases}-\dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{% \frac{1}{2}}2^{\nu+1}\Gamma\left(\frac{1}{2}-\nu\right)},&0<a<\|b-c\|,\\ 0,&\|b-c\|<a<b+c,\\ \dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{\frac{1}{2}}2^{\nu+1}\Gamma% \left(\frac{1}{2}-\nu\right)},&a>b+c.\end{cases}$
ⓘ 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, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\nu$ : complex parameter and $A$ : area Keywords: Mellin transform Referenced by: §10.22(iv), §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E75 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b).

§10.22(v) Hankel Transform

ⓘ

Keywords:: Bessel functions, Hankel transform, Hankel’s inversion theorem, Weber transform, integrals of Bessel and Hankel functions
Referenced by:: Ch.10, §3.5(ix), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/10.22.v
Addition (effective with 1.0.11):: A reference to Galapon and Martinez (2014) was added.
See also:: Annotations for §10.22 and Ch.10

The Hankel transform (or Bessel transform) of a function $f(x)$ is defined as

10.22.76		$g(y)=\int_{0}^{\infty}f(x)J_{\nu}\left(xy\right)(xy)^{\frac{1}{2}}\,\mathrm{d}x.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $y$ : real variable, $\nu$ : complex parameter, $f(x)$ : function and $g(y)$ : function Referenced by: §1.18(vi), §10.74(vii) Permalink: http://dlmf.nist.gov/10.22.E76 Encodings: TeX, pMML, png See also: Annotations for §10.22(v), §10.22 and Ch.10

Hankel’s inversion theorem is given by

10.22.77		$f(y)=\int_{0}^{\infty}g(x)J_{\nu}\left(xy\right)(xy)^{\frac{1}{2}}\,\mathrm{d}x.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $y$ : real variable, $\nu$ : complex parameter, $f(x)$ : function and $g(y)$ : function Referenced by: §1.18(vi), §10.22(v) Permalink: http://dlmf.nist.gov/10.22.E77 Encodings: TeX, pMML, png See also: Annotations for §10.22(v), §10.22 and Ch.10

Sufficient conditions for the validity of (10.22.77) are that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ when $\nu\geq-\tfrac{1}{2}$ , or that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ and $\int_{0}^{1}x^{\nu+\frac{1}{2}}|f(x)|\,\mathrm{d}x<\infty$ when $-1<\nu<-\tfrac{1}{2}$ ; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62).

For asymptotic expansions of Hankel transforms see Wong (1976, 1977), Frenzen and Wong (1985a) and Galapon and Martinez (2014).

For collections of Hankel transforms see Erdélyi et al. (1954b, Chapter 8) and Oberhettinger (1972).

The following two formulas are generalizations of the Hankel transform. These are examples of the self-adjoint extensions and the Weyl alternatives of §1.18(ix).

10.22.78		$f(x)=\int_{0}^{\infty}(xt)^{\frac{1}{2}}\frac{J_{\nu}\left(xt\right)Y_{\nu}% \left(at\right)-Y_{\nu}\left(xt\right)J_{\nu}\left(at\right)}{{J_{\nu}}^{2}% \left(at\right)+{Y_{\nu}}^{2}\left(at\right)}\*\int_{a}^{\infty}(yt)^{\frac{1}% {2}}\left(J_{\nu}\left(yt\right)Y_{\nu}\left(at\right)-Y_{\nu}\left(yt\right)J% _{\nu}\left(at\right)\right)f(y)\,\mathrm{d}y\,\mathrm{d}t,$
$a>0$ .
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $y$ : real variable, $\nu$ : complex parameter and $f(x)$ : function Sources: Titchmarsh (1962a, p. 87); Watson (1944, §14.52) Referenced by: §1.18(ix), §1.18(vi), §10.22(vi), Erratum (V1.2.0) for Chapter 10 Additions Permalink: http://dlmf.nist.gov/10.22.E78 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §10.22(v), §10.22 and Ch.10

This is the Weber transform. A sufficient condition for the validity is $\int_{a}^{\infty}|f(y)|\,\mathrm{d}y<\infty$ .

10.22.79		$f(x)=\int_{0}^{\infty}(xt)^{\frac{1}{2}}\frac{cJ_{\nu}\left(xt\right)+t^{2\nu}% J_{-\nu}\left(xt\right)}{c^{2}+2c\cos\left(\nu\pi\right)t^{2\nu}+t^{4\nu}}\*% \int_{0}^{\infty}(yt)^{\frac{1}{2}}\left(cJ_{\nu}\left(yt\right)+t^{2\nu}J_{-% \nu}\left(yt\right)\right)f(y)\,\mathrm{d}y\,\mathrm{d}t,$
$0<\nu<1,c>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $y$ : real variable, $\nu$ : complex parameter and $f(x)$ : function Referenced by: §1.18(ix), §1.18(vi), §10.22(v), §10.22(vi), Erratum (V1.2.0) for Chapter 10 Additions Permalink: http://dlmf.nist.gov/10.22.E79 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §10.22(v), §10.22 and Ch.10

Sufficient conditions for the validity of (10.22.79) are that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ when $0<\nu\leq\tfrac{1}{2}$ , or that $\int_{0}^{\infty}|f(x)|\,\mathrm{d}x<\infty$ and $\int_{0}^{1}x^{\frac{1}{2}-\nu}|f(x)|\,\mathrm{d}x<\infty$ when $\tfrac{1}{2}<\nu<1$ ; see Titchmarsh (1962a, pp. 88–90).

§10.22(vi) Compendia

ⓘ

Keywords:: compendia, integrals of Bessel and Hankel functions
Permalink:: http://dlmf.nist.gov/10.22.vi
Addition (effective with 1.2.0):: Equations (10.22.78), (10.22.79) were added.
See also:: Annotations for §10.22 and Ch.10

For collections of integrals of the functions $J_{\nu}\left(z\right)$ , $Y_{\nu}\left(z\right)$ , ${H^{(1)}_{\nu}}\left(z\right)$ , and ${H^{(2)}_{\nu}}\left(z\right)$ , including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2015, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).

10.21 Zeros 10.23 Sums

NIST

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§10.22 Integrals

Contents

§10.22(i) Indefinite Integrals

Products

§10.22(ii) Integrals over Finite Intervals

Trigonometric Arguments

Products

Convolutions

Fractional Integral

Orthogonality

§10.22(iii) Integrals over the Interval (x,∞)

§10.22(iv) Integrals over the Interval (0,∞)

Orthogonality

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

Other Double Products

Triple Products

§10.22(v) Hankel Transform

§10.22(vi) Compendia

§10.22(iii) Integrals over the Interval $(x,\infty)$

§10.22(iv) Integrals over the Interval $(0,\infty)$