§10.22 Integrals

Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/10.22

§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

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).
Keywords:: cylinder functions, integrals, integrals of Bessel and Hankel functions
Permalink:: http://dlmf.nist.gov/10.22.i

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

10.22.1

$\int z^{{\nu+1}}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)dz=z^{{\nu+1}}\mathop{\mathscr{C}_{{\nu+1}}\/}\nolimits\!\left(z\right),$

$\int z^{{-\nu+1}}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)dz=-z^{{-\nu+1}}\mathop{\mathscr{C}_{{\nu-1}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $dx$ : 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
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10.22.2 $\int z^{\nu}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)dz=\pi^{{\frac{1}{2}}}2^{{\nu-1}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)\* z\left(\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{\mathbf{H}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathscr{C}_{{\nu-1}}\/}\nolimits\!\left(z\right)\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $dx$ : 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
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For the Struve function $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ see §11.2(i).

10.22.3

$\int e^{{iz}}z^{\nu}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)dz=\frac{e^{{iz}}z^{{\nu+1}}}{2\nu+1}(\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)-i\mathop{\mathscr{C}_{{\nu+1}}\/}\nolimits\!\left(z\right)),$ $\nu\neq-\tfrac{1}{2}$ ,

$\int e^{{iz}}z^{{-\nu}}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)dz=\frac{e^{{iz}}z^{{-\nu+1}}}{1-2\nu}(\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)+i\mathop{\mathscr{C}_{{\nu-1}}\/}\nolimits\!\left(z\right)),$ $\nu\neq\tfrac{1}{2}$ .

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : 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
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¶ Products

Keywords:: cylinder functions, integrals, integrals of Bessel and Hankel functions

10.22.4 $\int z\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\mu}(bz)dz=\frac{z\left(a\mathop{\mathscr{C}_{{\mu+1}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\mu}(bz)-b\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{{\mu+1}}(bz)\right)}{a^{2}-b^{2}},$ $a^{2}\neq b^{2}$ ,

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $dx$ : 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

10.22.5 $\int z\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\mu}(az)dz=\tfrac{1}{4}z^{2}\left(2\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\mu}(az)-\mathop{\mathscr{C}_{{\mu-1}}\/}\nolimits\!\left(az\right)\mathscr{D}_{{\mu+1}}(az)-\mathop{\mathscr{C}_{{\mu+1}}\/}\nolimits\!\left(az\right)\mathscr{D}_{{\mu-1}}(az)\right),$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $dx$ : 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

10.22.6 $\int\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)\frac{dz}{z}=-\frac{az(\mathop{\mathscr{C}_{{\mu+1}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)-\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{{\nu+1}}(az))}{\mu^{2}-\nu^{2}}+\frac{\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)}{\mu+\nu},$ $\mu^{2}\neq\nu^{2}$ ,

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $dx$ : 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

10.22.7

$\int z^{{\mu+\nu+1}}\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)dz=\frac{z^{{\mu+\nu+2}}}{2(\mu+\nu+1)}\*\left(\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)+\mathop{\mathscr{C}_{{\mu+1}}\/}\nolimits\!\left(az\right)\mathscr{D}_{{\nu+1}}(az)\right),$ $\mu+\nu\neq-1$ ,

$\int z^{{-\mu-\nu+1}}\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)dz=\frac{z^{{-\mu-\nu+2}}}{2(1-\mu-\nu)}\*\left(\mathop{\mathscr{C}_{{\mu}}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)+\mathop{\mathscr{C}_{{\mu-1}}\/}\nolimits\!\left(az\right)\mathscr{D}_{{\nu-1}}(az)\right),$ $\mu+\nu\neq 1$ .

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $dx$ : 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
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§10.22(ii) Integrals 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 $\int _{0}^{{2\pi}}\mathop{J_{{2\nu}}\/}\nolimits\!\left(2z\mathop{\sin\/}\nolimits\theta\right)e^{{\pm 2i\mu\theta}}d\theta=\pi e^{{\pm i\mu\pi}}\mathop{J_{{\nu+\mu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu-\mu}}\/}\nolimits\!\left(z\right)$ , $\realpart{\nu}>-\tfrac{1}{2}$ , 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 $\int _{0}^{x}t^{{-1}}{\mathop{J_{{n}}\/}\nolimits^{{2}}}\!\left(t\right)dt-\tfrac{1}{2}{\mathop{J_{{n}}\/}\nolimits^{{2}}}\!\left(x\right)$ ; 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 $\mathop{\mathscr{C}_{{\mu\pm 1}}\/}\nolimits\!\left(az\right)$ and $\mathscr{D}_{{\mu\pm 1}}(bz)$ by $\mp{\mathop{\mathscr{C}_{{\mu}}\/}\nolimits^{{\prime}}}\!\left(az\right)$ and $\mp\mathscr{D}_{{\mu}}^{{\prime}}(bz)$ , respectively, by means of (10.6.2).
Keywords:: integrals of Bessel and Hankel functions
Permalink:: http://dlmf.nist.gov/10.22.ii

Throughout this subsection $x>0$ .

10.22.8 $\int _{0}^{x}\mathop{J_{{\nu}}\/}\nolimits\!\left(t\right)dt=2\sum _{{k=0}}^{\infty}\mathop{J_{{\nu+2k+1}}\/}\nolimits\!\left(x\right),$ $\realpart{\nu}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.9 $\int _{0}^{x}\mathop{J_{{2n}}\/}\nolimits\!\left(t\right)dt=\int _{0}^{x}\mathop{J_{{0}}\/}\nolimits\!\left(t\right)dt-2\sum _{{k=0}}^{{n-1}}\mathop{J_{{2k+1}}\/}\nolimits\!\left(x\right),\quad\int _{0}^{x}\mathop{J_{{2n+1}}\/}\nolimits\!\left(t\right)dt=1-\mathop{J_{{0}}\/}\nolimits\!\left(x\right)-2\sum _{{k=1}}^{n}\mathop{J_{{2k}}\/}\nolimits\!\left(x\right),$ $n=0,1,\dots$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : 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

10.22.10 $\int _{0}^{x}t^{\mu}\mathop{J_{{\nu}}\/}\nolimits\!\left(t\right)dt=x^{\mu}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}\*\sum _{{k=0}}^{\infty}\frac{(\nu+2k+1)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}\mathop{J_{{\nu+2k+1}}\/}\nolimits\!\left(x\right),$ $\realpart{(\mu+\nu+1)}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.11 $\int _{0}^{x}\frac{1-\mathop{J_{{0}}\/}\nolimits\!\left(t\right)}{t}dt=\frac{1}{2}\sum _{{k=1}}^{\infty}\frac{\mathop{\psi\/}\nolimits\!\left(k+1\right)-\mathop{\psi\/}\nolimits\!\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}\mathop{J_{{k}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\int$ : integral, $k$ : nonnegative integer and $x$ : real variable
Referenced by:: §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.22.E11
Encodings:: TeX, pMML, png

10.22.12 $x\int _{0}^{x}\frac{1-\mathop{J_{{0}}\/}\nolimits\!\left(t\right)}{t}dt=2\sum _{{k=0}}^{\infty}(2k+3)(\mathop{\psi\/}\nolimits\!\left(k+2\right)-\mathop{\psi\/}\nolimits\!\left(1\right))\mathop{J_{{2k+3}}\/}\nolimits\!\left(x\right)=x-2\!\mathop{J_{{1}}\/}\nolimits\!\left(x\right)+2\sum _{{k=0}}^{\infty}(2k+5)\*(\mathop{\psi\/}\nolimits\!\left(k+3\right)-\mathop{\psi\/}\nolimits\!\left(1\right)-1)\mathop{J_{{2k+5}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $k$ : nonnegative integer and $x$ : real variable
A&S Ref:: 11.1.19
Referenced by:: §10.22(ii), §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.22.E12
Encodings:: TeX, pMML, png

where $\mathop{\psi\/}\nolimits\!\left(x\right)={\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$ (§5.2(i)). See also (10.22.39).

¶ Trigonometric Arguments

Keywords:: integrals of Bessel and Hankel functions

10.22.13 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{2\nu}}\/}\nolimits\!\left(2z\mathop{\cos\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits\!\left(2\mu\theta\right)d\theta=\tfrac{1}{2}\pi\mathop{J_{{\nu+\mu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu-\mu}}\/}\nolimits\!\left(z\right),$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.14 $\int _{0}^{\pi}\mathop{J_{{2\nu}}\/}\nolimits\!\left(2z\mathop{\sin\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits\!\left(2\mu\theta\right)d\theta=\pi\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)\mathop{J_{{\nu+\mu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu-\mu}}\/}\nolimits\!\left(z\right),$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.22.15 $\int _{0}^{\pi}\mathop{J_{{2\nu}}\/}\nolimits\!\left(2z\mathop{\sin\/}\nolimits\theta\right)\mathop{\sin\/}\nolimits(2\mu\theta)d\theta=\pi\mathop{\sin\/}\nolimits(\mu\pi)\mathop{J_{{\nu+\mu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu-\mu}}\/}\nolimits\!\left(z\right),$ $\realpart{\nu}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.22.16 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{0}}\/}\nolimits\!\left(2z\mathop{\sin\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits(2n\theta)d\theta=\tfrac{1}{2}\pi{\mathop{J_{{n}}\/}\nolimits^{{2}}}\!\left(z\right),$ $n=0,1,2,\ldots$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

10.22.17 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{Y_{{2\nu}}\/}\nolimits\!\left(2z\mathop{\cos\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits(2\mu\theta)d\theta=\tfrac{1}{2}\pi\mathop{\cot\/}\nolimits(2\nu\pi)\mathop{J_{{\nu+\mu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu-\mu}}\/}\nolimits\!\left(z\right)-\tfrac{1}{2}\pi\mathop{\csc\/}\nolimits(2\nu\pi)\mathop{J_{{\mu-\nu}}\/}\nolimits\!\left(z\right)\mathop{J_{{-\mu-\nu}}\/}\nolimits\!\left(z\right),$ $-\tfrac{1}{2}<\realpart{\nu}<\tfrac{1}{2}$ ,

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, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.18 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{Y_{{0}}\/}\nolimits\!\left(2z\mathop{\sin\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits\!\left(2n\theta\right)d\theta=\tfrac{1}{2}\pi\mathop{J_{{n}}\/}\nolimits\!\left(z\right)\mathop{Y_{{n}}\/}\nolimits\!\left(z\right),$ $n=0,1,2,\dots$ .

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, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

10.22.19 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{\mu}}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{{\mu+1}}(\mathop{\cos\/}\nolimits\theta)^{{2\nu+1}}d\theta=2^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)z^{{-\nu-1}}\mathop{J_{{\mu+\nu+1}}\/}\nolimits\!\left(z\right),$ $\realpart{\mu}>-1$ , $\realpart{\nu}>-1$ ,

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, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.22.20 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{\mu}}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{\mu}(\mathop{\cos\/}\nolimits\theta)^{{2\mu}}d\theta=\pi^{{\frac{1}{2}}}2^{{\mu-1}}z^{{-\mu}}\*\mathop{\Gamma\/}\nolimits\!\left(\mu+\tfrac{1}{2}\right){\mathop{J_{{\mu}}\/}\nolimits^{{2}}}\!\left(\tfrac{1}{2}z\right),$ $\realpart{\mu}>-\tfrac{1}{2}$ ,

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, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E20
Encodings:: TeX, pMML, png

10.22.21 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{Y_{{\mu}}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{\mu}(\mathop{\cos\/}\nolimits\theta)^{{2\mu}}d\theta=\pi^{{\frac{1}{2}}}2^{{\mu-1}}z^{{-\mu}}\*\mathop{\Gamma\/}\nolimits\!\left(\mu+\tfrac{1}{2}\right)\mathop{J_{{\mu}}\/}\nolimits\!\left(\tfrac{1}{2}z\right)\mathop{Y_{{\mu}}\/}\nolimits\!\left(\tfrac{1}{2}z\right),$ $\realpart{\mu}>-\tfrac{1}{2}$ .

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, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E21
Encodings:: TeX, pMML, png

10.22.22 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{\mu}}\/}\nolimits\!\left(z{\mathop{\sin\/}\nolimits^{{2}}}\theta\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(z{\mathop{\cos\/}\nolimits^{{2}}}\theta\right)(\mathop{\sin\/}\nolimits\theta)^{{2\mu+1}}(\mathop{\cos\/}\nolimits\theta)^{{2\nu+1}}d\theta=\frac{\mathop{\Gamma\/}\nolimits\!\left(\mu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)\mathop{J_{{\mu+\nu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)}{(8\pi z)^{\frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu+1\right)},$ $\realpart{\mu}>-\tfrac{1}{2},\realpart{\nu}>-\tfrac{1}{2}$ .

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, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.22.23 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{\mu}}\/}\nolimits\!\left(z{\mathop{\sin\/}\nolimits^{{2}}}\theta\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(z{\mathop{\cos\/}\nolimits^{{2}}}\theta\right)(\mathop{\sin\/}\nolimits\theta)^{{2\alpha-1}}\mathop{\sec\/}\nolimits\theta d\theta=\frac{(\mu+\nu+\alpha)\mathop{\Gamma\/}\nolimits\!\left(\mu+\alpha\right)2^{{\alpha-1}}}{\nu\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)z^{\alpha}}\mathop{J_{{\mu+\nu+\alpha}}\/}\nolimits\!\left(z\right),$ $\realpart{(\mu+\alpha)}>0$ , $\realpart{\nu}>0$ .

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, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 11.4.11
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E23
Encodings:: TeX, pMML, png

10.22.24 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{\mu}}\/}\nolimits\!\left(z{\mathop{\sin\/}\nolimits^{{2}}}\theta\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(z{\mathop{\cos\/}\nolimits^{{2}}}\theta\right)\mathop{\cot\/}\nolimits\theta d\theta=\tfrac{1}{2}\mu^{{-1}}\mathop{J_{{\mu+\nu}}\/}\nolimits\!\left(z\right),$ $\realpart{\mu}>0,\realpart{\nu}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.22.25 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{\mu}}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)\mathop{I_{{\nu}}\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)(\mathop{\tan\/}\nolimits\theta)^{{\mu+1}}d\theta=\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{1}{2}z)^{\mu}}{2\!\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right),$ $\realpart{\nu}>\realpart{\mu}>-1$ .

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, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : 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

For $\mathop{I_{{\nu}}\/}\nolimits$ see §10.25(ii).

10.22.26 $\int _{0}^{{\frac{1}{2}\pi}}\mathop{J_{{\mu}}\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(\zeta\mathop{\cos\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{{\mu+1}}(\mathop{\cos\/}\nolimits\theta)^{{\nu+1}}d\theta=\frac{z^{\mu}\zeta^{\nu}\mathop{J_{{\mu+\nu+1}}\/}\nolimits\!\left(\sqrt{\zeta^{2}+z^{2}}\right)}{(\zeta^{2}+z^{2})^{{\frac{1}{2}(\mu+\nu+1)}}},$ $\realpart{\mu}>-1,\realpart{\nu}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

¶ Products

10.22.27 $\int _{0}^{x}t{\mathop{J_{{\nu-1}}\/}\nolimits^{{2}}}\!\left(t\right)dt=2\sum _{{k=0}}^{\infty}(\nu+2k){\mathop{J_{{\nu+2k}}\/}\nolimits^{{2}}}\!\left(x\right),$ $\realpart{\nu}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.28 $\int _{0}^{x}t\left({\mathop{J_{{\nu-1}}\/}\nolimits^{{2}}}\!\left(t\right)-{\mathop{J_{{\nu+1}}\/}\nolimits^{{2}}}\!\left(t\right)\right)dt=2\nu{\mathop{J_{{\nu}}\/}\nolimits^{{2}}}\!\left(x\right),$ $\realpart{\nu}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.29 $\int _{0}^{x}t{\mathop{J_{{0}}\/}\nolimits^{{2}}}\!\left(t\right)dt=\tfrac{1}{2}x^{2}\left({\mathop{J_{{0}}\/}\nolimits^{{2}}}\!\left(x\right)+{\mathop{J_{{1}}\/}\nolimits^{{2}}}\!\left(x\right)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : 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

10.22.30 $\int _{0}^{x}\mathop{J_{{n}}\/}\nolimits\!\left(t\right)\mathop{J_{{n+1}}\/}\nolimits\!\left(t\right)dt=\tfrac{1}{2}\left(1-{\mathop{J_{{0}}\/}\nolimits^{{2}}}\!\left(x\right)\right)-\sum _{{k=1}}^{n}{\mathop{J_{{k}}\/}\nolimits^{{2}}}\!\left(x\right)=\sum _{{k=n+1}}^{\infty}{\mathop{J_{{k}}\/}\nolimits^{{2}}}\!\left(x\right),$ $n=0,1,2,\ldots$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : 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

¶ Convolutions

Keywords:: integrals of Bessel and Hankel functions

10.22.31 $\int _{0}^{x}\mathop{J_{{\mu}}\/}\nolimits\!\left(t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(x-t\right)dt=2\sum _{{k=0}}^{\infty}(-1)^{k}\mathop{J_{{\mu+\nu+2k+1}}\/}\nolimits\!\left(x\right),$ $\realpart{\mu}>-1,\realpart{\nu}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.32 $\int _{0}^{x}\mathop{J_{{\nu}}\/}\nolimits\!\left(t\right)\mathop{J_{{1-\nu}}\/}\nolimits\!\left(x-t\right)dt=\mathop{J_{{0}}\/}\nolimits\!\left(x\right)-\mathop{\cos\/}\nolimits x,$ $-1<\realpart{\nu}<2$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.33 $\int _{0}^{x}\mathop{J_{{\nu}}\/}\nolimits\!\left(t\right)\mathop{J_{{-\nu}}\/}\nolimits\!\left(x-t\right)dt=\mathop{\sin\/}\nolimits x,$ $|\realpart{\nu}|<1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.22.34 $\int _{0}^{x}t^{{-1}}\mathop{J_{{\mu}}\/}\nolimits\!\left(t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(x-t\right)dt=\frac{\mathop{J_{{\mu+\nu}}\/}\nolimits\!\left(x\right)}{\mu},$ $\realpart{\mu}>0,\realpart{\nu}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.35 $\int _{0}^{x}\frac{\mathop{J_{{\mu}}\/}\nolimits\!\left(t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(x-t\right)dt}{t(x-t)}=\frac{(\mu+\nu)\mathop{J_{{\mu+\nu}}\/}\nolimits\!\left(x\right)}{\mu\nu x},$ $\realpart{\mu}>0,\realpart{\nu}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

¶ Fractional Integral

Keywords:: integrals of Bessel and Hankel functions

10.22.36 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)}\int _{0}^{x}(x-t)^{{\alpha-1}}\mathop{J_{{\nu}}\/}\nolimits\!\left(t\right)dt=2^{\alpha}\sum _{{k=0}}^{\infty}\frac{(\alpha)_{k}}{k!}\mathop{J_{{\nu+\alpha+2k}}\/}\nolimits\!\left(x\right),$ $\realpart{\alpha}>0,\realpart{\nu}\geq 0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\realpart{}$ : 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

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

¶ Orthogonality

Keywords:: Bessel functions, integrals of Bessel and Hankel functions

If $\nu>-1$ , then

10.22.37 $\int _{0}^{1}t\mathop{J_{{\nu}}\/}\nolimits\!\left(j_{{\nu,\ell}}t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(j_{{\nu,m}}t\right)dt=\tfrac{1}{2}\delta _{{\ell,m}}\left({\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(j_{{\nu,\ell}}\right)\right)^{2},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\delta _{{j,k}}$ : Kronecker delta, $dx$ : differential of $x$ , $\int$ : integral, $m$ : integer and $\nu$ : complex parameter
A&S Ref:: 11.4.5
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E37
Encodings:: TeX, pMML, png

where $j_{{\nu,\ell}}$ and $j_{{\nu,m}}$ are zeros of $\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)$ (§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 $\int _{0}^{1}t\mathop{J_{{\nu}}\/}\nolimits\!\left(\alpha _{\ell}t\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(\alpha _{m}t\right)dt=\delta _{{\ell,m}}\left(\frac{a^{2}}{b^{2}}+\alpha _{\ell}^{2}-\nu^{2}\right)\frac{(\mathop{J_{{\nu}}\/}\nolimits\!\left(\alpha _{\ell}\right))^{2}}{2\alpha _{\ell}^{2}},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\delta _{{j,k}}$ : Kronecker delta, $dx$ : differential of $x$ , $\int$ : integral, $m$ : integer and $\nu$ : complex parameter
A&S Ref:: 11.4.5
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E38
Encodings:: TeX, pMML, png

where $\alpha _{\ell}$ and $\alpha _{m}$ are positive zeros of $a\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)+bx{\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right)$ . (Compare (10.22.55)).

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

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.
Keywords:: integrals of Bessel and Hankel functions
Permalink:: http://dlmf.nist.gov/10.22.iii

When $x>0$

10.22.39 $\int _{x}^{\infty}\frac{\mathop{J_{{0}}\/}\nolimits\!\left(t\right)}{t}dt+\EulerConstant+\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)=\int _{0}^{x}\frac{1-\mathop{J_{{0}}\/}\nolimits\!\left(t\right)}{t}dt=\sum _{{k=1}}^{\infty}(-1)^{{k-1}}\frac{(\frac{1}{2}x)^{{2k}}}{2k(k!)^{2}},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\EulerConstant$ : Euler’s constant, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

10.22.40 $\int _{x}^{\infty}\frac{\mathop{Y_{{0}}\/}\nolimits\!\left(t\right)}{t}dt=-\frac{1}{\pi}\left(\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)+\EulerConstant\right)^{2}+\frac{\pi}{6}+\frac{2}{\pi}\sum _{{k=1}}^{\infty}(-1)^{k}\*\left(\mathop{\psi\/}\nolimits\!\left(k+1\right)+\frac{1}{2k}-\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{{2k}}}{2k(k!)^{2}},$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\EulerConstant$ : Euler’s constant, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : 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

where $\EulerConstant$ is Euler’s constant (§5.2(ii)).

§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, pp. 411) and Askey et al. (1986).
Permalink:: http://dlmf.nist.gov/10.22.iv

10.22.41 $\int _{0}^{\infty}\mathop{J_{{\nu}}\/}\nolimits\!\left(t\right)dt=1,$ $\realpart{\nu}>-1$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

10.22.42 $\int _{0}^{\infty}\mathop{Y_{{\nu}}\/}\nolimits\!\left(t\right)dt=-\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\nu\pi\right),$ $|\realpart{\nu}|<1$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\tan\/}\nolimits z$ : 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

10.22.43 $\int _{0}^{\infty}t^{\mu}\mathop{J_{{\nu}}\/}\nolimits\!\left(t\right)dt=2^{\mu}\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},$ $\realpart{(\mu+\nu)}>-1$ , $\realpart{\mu}<\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part and $\nu$ : complex parameter
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

10.22.44 $\int _{0}^{\infty}t^{\mu}\mathop{Y_{{\nu}}\/}\nolimits\!\left(t\right)dt=\frac{2^{\mu}}{\pi}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi,$ $\realpart{(\mu\pm\nu)}>-1$ , $\realpart{\mu}<\tfrac{1}{2}$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function and $\nu$ : complex parameter
A&S Ref:: 11.4.19 (modified)
Referenced by:: §10.22(iii)
Permalink:: http://dlmf.nist.gov/10.22.E44
Encodings:: TeX, pMML, png

10.22.45 $\int _{0}^{\infty}\frac{1-\mathop{J_{{0}}\/}\nolimits\!\left(t\right)}{t^{\mu}}dt=-\frac{\pi\mathop{\sec\/}\nolimits\!\left(\frac{1}{2}\mu\pi\right)}{2^{\mu}{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(\frac{1}{2}\mu+\frac{1}{2}\right)},$ $1<\realpart{\mu}<3$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part and $\mathop{\sec\/}\nolimits z$ : secant function
A&S Ref:: 11.4.18 (modified)
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E45
Encodings:: TeX, pMML, png

10.22.46 $\int _{0}^{\infty}\frac{t^{{\nu+1}}\mathop{J_{{\nu}}\/}\nolimits\!\left(at\right)}{(t^{2}+b^{2})^{{\mu+1}}}dt=\frac{a^{\mu}b^{{\nu-\mu}}}{2^{\mu}\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)}\mathop{K_{{\nu-\mu}}\/}\nolimits\!\left(ab\right),$ $a>0$ , $\realpart{b}>0$ , $-1<\realpart{\nu}<2\realpart{\mu}+\tfrac{3}{2}$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
A&S Ref:: 11.4.44
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E46
Encodings:: TeX, pMML, png

10.22.47 $\int _{0}^{\infty}\frac{t^{\nu}\mathop{Y_{{\nu}}\/}\nolimits\!\left(at\right)}{t^{2}+b^{2}}dt=-b^{{\nu-1}}\mathop{K_{{\nu}}\/}\nolimits\!\left(ab\right),$ $a>0,\realpart{b}>0,-\tfrac{1}{2}<\realpart{\nu}<\tfrac{5}{2}$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
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

For $\mathop{K_{{\nu}}\/}\nolimits$ see §10.25(ii).

10.22.48 $\int _{0}^{\infty}\mathop{J_{{\mu}}\/}\nolimits\!\left(x\mathop{\cosh\/}\nolimits\phi\right)(\mathop{\cosh\/}\nolimits\phi)^{{1-\mu}}(\mathop{\sinh\/}\nolimits\phi)^{{2\nu+1}}d\phi=2^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)x^{{-\nu-1}}\mathop{J_{{\mu-\nu-1}}\/}\nolimits\!\left(x\right),$ $x>0,\realpart{\nu}>-1,\realpart{\mu}>2\realpart{\nu}+\tfrac{1}{2}$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : 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

10.22.49 $\int _{0}^{\infty}t^{{\mu-1}}e^{{-at}}\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)dt=\frac{(\tfrac{1}{2}b)^{\nu}}{a^{{\mu+\nu}}}\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu\right)\*\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),$ $\realpart{(\mu+\nu)}>0,\realpart{(a\pm ib)}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function and $\nu$ : complex parameter
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E49
Encodings:: TeX, pMML, png

10.22.50 $\int _{0}^{\infty}t^{{\mu-1}}e^{{-at}}\mathop{Y_{{\nu}}\/}\nolimits\!\left(bt\right)dt=\mathop{\cot\/}\nolimits(\nu\pi)\frac{(\tfrac{1}{2}b)^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu\right)}{(a^{2}+b^{2})^{{\frac{1}{2}(\mu+\nu)}}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{\mu+\nu}{2},\frac{1-\mu+\nu}{2};\nu+1;\frac{b^{2}}{a^{2}+b^{2}}\right)-\mathop{\csc\/}\nolimits(\nu\pi)\frac{(\tfrac{1}{2}b)^{{-\nu}}\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)}{(a^{2}+b^{2})^{{\frac{1}{2}(\mu-\nu)}}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{\mu-\nu}{2},\frac{1-\mu-\nu}{2};1-\nu;\frac{b^{2}}{a^{2}+b^{2}}\right),$ $\realpart{\mu}>|\realpart{\nu}|,\realpart{(a\pm ib)}>0$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.22.E50
Encodings:: TeX, pMML, png

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

10.22.51 $\int _{0}^{\infty}\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}\nolimits\!\left(-p^{2}t^{2}\right)t^{{\nu+1}}dt=\frac{b^{\nu}}{(2p^{2})^{{\nu+1}}}\mathop{\exp\/}\nolimits\!\left(-\frac{b^{2}}{4p^{2}}\right),$ $\realpart{\nu}>-1$ , $\realpart{(p^{2})}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\realpart{}$ : real part and $\nu$ : complex parameter
A&S Ref:: 11.4.29
Permalink:: http://dlmf.nist.gov/10.22.E51
Encodings:: TeX, pMML, png

10.22.52 $\int _{0}^{\infty}\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}\nolimits(-p^{2}t^{2})dt=\frac{\sqrt{\pi}}{2p}\mathop{\exp\/}\nolimits\left(-\frac{b^{2}}{8p^{2}}\right)\mathop{I_{{\ifrac{\nu}{2}}}\/}\nolimits\left(\frac{b^{2}}{8p^{2}}\right),$ $\realpart{\nu}>-1,\realpart{(p^{2})}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.22.E52
Encodings:: TeX, pMML, png

10.22.53 $\int _{0}^{\infty}\mathop{Y_{{2\nu}}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}\nolimits\!\left(-p^{2}t^{2}\right)dt=-\frac{\sqrt{\pi}}{2p}\mathop{\exp\/}\nolimits\!\left(-\frac{b^{2}}{8p^{2}}\right)\left(\mathop{I_{{\nu}}\/}\nolimits\left(\frac{b^{2}}{8p^{2}}\right)\mathop{\tan\/}\nolimits\!\left(\nu\pi\right)+\frac{1}{\pi}\mathop{K_{{\nu}}\/}\nolimits\left(\frac{b^{2}}{8p^{2}}\right)\mathop{\sec\/}\nolimits\!\left(\nu\pi\right)\right),$ $|\realpart{\nu}|<\tfrac{1}{2}$ , $\realpart{(p^{2})}>0$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\tan\/}\nolimits z$ : tangent function and $\nu$ : complex parameter
A&S Ref:: 11.4.30
Permalink:: http://dlmf.nist.gov/10.22.E53
Encodings:: TeX, pMML, png

For $\mathop{I\/}\nolimits$ and $\mathop{K\/}\nolimits$ see §10.25(ii).

10.22.54 $\int _{0}^{\infty}\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}\nolimits\!\left(-p^{2}t^{2}\right)t^{{\mu-1}}dt=\frac{(\tfrac{1}{2}b/p)^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu\right)}{2p^{\mu}}\mathop{\exp\/}\nolimits\!\left(-\frac{b^{2}}{4p^{2}}\right)\*\mathop{{\mathbf{M}}\/}\nolimits\!\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1,\nu+1,\frac{b^{2}}{4p^{2}}\right),$ $\realpart{(\mu+\nu)}>0$ , $\realpart{(p^{2})}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\realpart{}$ : 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

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

¶ Orthogonality

Keywords:: Bessel functions, integrals of Bessel and Hankel functions

10.22.55 $\int _{0}^{\infty}t^{{-1}}\mathop{J_{{\nu+2\ell+1}}\/}\nolimits\!\left(t\right)\mathop{J_{{\nu+2m+1}}\/}\nolimits\!\left(t\right)dt=\frac{\delta _{{\ell,m}}}{2(2\ell+\nu+1)},$ $\nu+\ell+m>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\delta _{{j,k}}$ : Kronecker delta, $dx$ : 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

¶ Weber–Schafheitlin Discontinuous Integrals, including Special Cases

Keywords:: Bessel functions, Weber–Schafheitlin discontinuous integrals

10.22.56 $\int _{0}^{\infty}\frac{\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)}{t^{\lambda}}dt=\frac{a^{\mu}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu-\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}b^{{\mu-\lambda+1}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\lambda+\frac{1}{2}\right)}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}(\mu+\nu-\lambda+1),\tfrac{1}{2}(\mu-\nu-\lambda+1);\mu+1;\frac{a^{2}}{b^{2}}\right),$ $0<a<b$ , $\realpart{(\mu+\nu+1)}>\realpart{\lambda}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function and $\nu$ : complex parameter
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

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

10.22.57 $\int _{0}^{\infty}\frac{\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(at\right)}{t^{\lambda}}dt=\frac{(\frac{1}{2}a)^{{\lambda-1}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\mu+\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\lambda\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\lambda+\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\lambda+\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\nu+\frac{1}{2}\right)},$ $\realpart{(\mu+\nu+1)}>\realpart{\lambda}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.22.E57
Encodings:: TeX, pMML, png

10.22.58 $\int _{0}^{\infty}\frac{\mathop{J_{{\nu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)}{t^{\lambda}}dt=\frac{(ab)^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu-\frac{1}{2}\lambda+\frac{1}{2}\right)}{2^{\lambda}(a^{2}+b^{2})^{{\nu-\frac{1}{2}\lambda+\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\lambda+\frac{1}{2}\right)}\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{2\nu+1-\lambda}{4},\frac{2\nu+3-\lambda}{4};\nu+1;\frac{4a^{2}b^{2}}{(a^{2}+b^{2})^{2}}\right),$ $a\neq b$ , $\realpart{(2\nu+1)}>\realpart{\lambda}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.22.E58
Encodings:: TeX, pMML, png

When $\realpart{\mu}>-1$

10.22.59 $\int _{0}^{\infty}e^{{ibt}}\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)dt=\begin{cases}\dfrac{\mathop{\exp\/}\nolimits\!\left(i\mu\mathop{\mathrm{arcsin}\/}\nolimits\!\left(b/a\right)\right)}{(a^{2}-b^{2})^{{\frac{1}{2}}}},&0\leq b<a,\\ \dfrac{ia^{\mu}\mathop{\exp\/}\nolimits\!\left(\frac{1}{2}\mu\pi i\right)}{(b^{2}-a^{2})^{{\frac{1}{2}}}\left(b+(b^{2}-a^{2})^{{\frac{1}{2}}}\right)^{\mu}},&0<a<b.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $\int$ : integral and $\mathop{\mathrm{arcsin}\/}\nolimits z$ : 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

10.22.60 $\int _{0}^{\infty}e^{{ibt}}\mathop{Y_{{0}}\/}\nolimits\!\left(at\right)dt=\begin{cases}(2i/\pi)(a^{2}-b^{2})^{{-\frac{1}{2}}}\mathop{\mathrm{arcsin}\/}\nolimits\!\left(b/a\right),&0\leq b<a,\\ (b^{2}-a^{2})^{{-\frac{1}{2}}}\left(-1+\dfrac{2i}{\pi}\mathop{\ln\/}\nolimits\!\left(\dfrac{a}{b+(b^{2}-a^{2})^{{\frac{1}{2}}}}\right)\right),&0<a<b.\end{cases}$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and $\mathop{\ln\/}\nolimits z$ : 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

When $\realpart{\mu}>0$ ,

10.22.61 $\int _{0}^{\infty}t^{{-1}}e^{{ibt}}\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)dt=\begin{cases}(1/\mu)\mathop{\exp\/}\nolimits\!\left(i\mu\mathop{\mathrm{arcsin}\/}\nolimits\!\left(b/a\right)\right),&0\leq b\leq a,\\ \dfrac{a^{\mu}\mathop{\exp\/}\nolimits\!\left(\frac{1}{2}\mu\pi i\right)}{\mu\left(b+(b^{2}-a^{2})^{{\frac{1}{2}}}\right)^{\mu}},&0<a\leq b.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $\int$ : integral and $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function
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

When $\realpart{\nu}>\realpart{\mu}>-1$ ,

10.22.62 $\int _{0}^{\infty}t^{{\mu-\nu+1}}\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)dt=\begin{cases}0,&0<b<a,\\ \dfrac{2^{{\mu-\nu+1}}a^{\mu}(b^{2}-a^{2})^{{\nu-\mu-1}}}{b^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu\right)},&0<a\leq b.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral and $\nu$ : complex parameter
A&S Ref:: 11.4.41
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E62
Encodings:: TeX, pMML, png

When $\realpart{\mu}>0$ ,

10.22.63 $\int _{0}^{\infty}\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\mu-1}}\/}\nolimits\!\left(bt\right)dt=\begin{cases}b^{{\mu-1}}a^{{-\mu}},&0<b<a,\\ (2b)^{{-1}},&b=a(>0),\\ 0,&0<a<b.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : 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

When $n=0,1,2,\dots$ and $\realpart{\mu}>-n-1$ ,

10.22.64 $\int _{0}^{\infty}\mathop{J_{{\mu+2n+1}}\/}\nolimits\!\left(at\right)\mathop{J_{{\mu}}\/}\nolimits\!\left(bt\right)dt=\begin{cases}\dfrac{b^{\mu}\mathop{\Gamma\/}\nolimits\!\left(\mu+n+1\right)}{a^{{\mu+1}}n!}\mathop{\mathbf{F}\/}\nolimits\!\left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}\right),&0<b<a,\\ (-1)^{n}/(2a),&b=a(>0),\\ 0,&0<a<b.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

10.22.65 $\int _{0}^{\infty}\mathop{J_{{0}}\/}\nolimits\!\left(at\right)\left(\mathop{J_{{0}}\/}\nolimits\!\left(bt\right)-\mathop{J_{{0}}\/}\nolimits\!\left(ct\right)\right)\frac{dt}{t}=\begin{cases}0,&0\leq b<a,0<c\leq a,\\ \mathop{\ln\/}\nolimits\!\left(c/a\right),&0\leq b<a\leq c.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\ln\/}\nolimits z$ : 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

¶ Other Double Products

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

10.22.66 $\int _{0}^{\infty}e^{{-at}}\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(ct\right)dt=\frac{1}{\pi(bc)^{{\frac{1}{2}}}}\*\mathop{Q_{{\nu-\frac{1}{2}}}\/}\nolimits\left(\frac{a^{2}+b^{2}+c^{2}}{2bc}\right),$ $\realpart{\nu}>-\tfrac{1}{2}$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $\nu$ : complex parameter
Referenced by:: ¶ ‣ §10.22(iv), §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E66
Encodings:: TeX, pMML, png

10.22.67 $\int _{0}^{\infty}t\mathop{\exp\/}\nolimits(-p^{2}t^{2})\mathop{J_{{\nu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)dt=\frac{1}{2p^{2}}\mathop{\exp\/}\nolimits\left(-\frac{a^{2}+b^{2}}{4p^{2}}\right)\mathop{I_{{\nu}}\/}\nolimits\left(\frac{ab}{2p^{2}}\right),$ $\realpart{\nu}>-1,\realpart{(p^{2})}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part and $\nu$ : complex parameter
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E67
Encodings:: TeX, pMML, png

10.22.68 $\int _{0}^{\infty}t\mathop{\exp\/}\nolimits(-p^{2}t^{2})\mathop{J_{{0}}\/}\nolimits\!\left(at\right)\mathop{Y_{{0}}\/}\nolimits\!\left(at\right)dt=-\frac{1}{2\pi p^{2}}\mathop{\exp\/}\nolimits\left(-\frac{a^{2}}{2p^{2}}\right)\mathop{K_{{0}}\/}\nolimits\left(\frac{a^{2}}{2p^{2}}\right),$ $\realpart{(p^{2})}>0$ .

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, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and $\realpart{}$ : real part
Referenced by:: §10.22(iv), §10.43(iv)
Permalink:: http://dlmf.nist.gov/10.22.E68
Encodings:: TeX, pMML, png

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

10.22.69 $\int _{0}^{\infty}\mathop{J_{{\nu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\frac{tdt}{t^{2}-z^{2}}=\left\{\begin{array}[]{ll}\frac{1}{2}\pi i\mathop{J_{{\nu}}\/}\nolimits\!\left(bz\right)\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(az\right),&a>b\\ \frac{1}{2}\pi i\mathop{J_{{\nu}}\/}\nolimits\!\left(az\right)\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(bz\right),&b>a\end{array}\right\},$ $\realpart{\nu}>-1,\imagpart{z}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $\imagpart{}$ : imaginary part, $\int$ : integral, $\realpart{}$ : real part, $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E69
Encodings:: TeX, pMML, png

10.22.70 $\int _{0}^{\infty}\mathop{Y_{{\nu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu+1}}\/}\nolimits\!\left(bt\right)\frac{tdt}{t^{2}-z^{2}}=\frac{1}{2}\pi\mathop{J_{{\nu+1}}\/}\nolimits\!\left(bz\right)\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(az\right),$ $a\geq b>0$ , $\realpart{\nu}>-\tfrac{3}{2},\imagpart{z}>0$ .

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, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $\imagpart{}$ : imaginary part, $\int$ : integral, $\realpart{}$ : 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

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

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

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

10.22.71 $\int _{0}^{\infty}\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(ct\right)t^{{1-\mu}}dt=\frac{(bc)^{{\mu-1}}(\mathop{\sin\/}\nolimits\phi)^{{\mu-\frac{1}{2}}}}{(2\pi)^{{\frac{1}{2}}}a^{\mu}}\mathop{\mathsf{P}^{{\frac{1}{2}-\mu}}_{{\nu-\frac{1}{2}}}\/}\nolimits(\mathop{\cos\/}\nolimits\phi),$ $\realpart{\mu}>-\tfrac{1}{2},\realpart{\nu}>-1,|b-c|<a<b+c,\mathop{\cos\/}\nolimits\phi=(b^{2}+c^{2}-a^{2})/(2bc)$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function and $\nu$ : complex parameter
Referenced by:: ¶ ‣ §10.22(iv), §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E71
Encodings:: TeX, pMML, png

10.22.72 $\int _{0}^{\infty}\mathop{J_{{\mu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(ct\right)t^{{1-\mu}}dt=\frac{(bc)^{{\mu-1}}\mathop{\cos\/}\nolimits(\nu\pi)(\mathop{\sinh\/}\nolimits\chi)^{{\mu-\frac{1}{2}}}}{(\frac{1}{2}\pi^{3})^{{\frac{1}{2}}}a^{\mu}}\mathop{Q^{{\frac{1}{2}-\mu}}_{{\nu-\frac{1}{2}}}\/}\nolimits(\mathop{\cosh\/}\nolimits\chi),$ $\realpart{\mu}>-\tfrac{1}{2},\realpart{\nu}>-1,a>b+c,\mathop{\cosh\/}\nolimits\chi=(a^{2}-b^{2}-c^{2})/(2bc)$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part and $\nu$ : complex parameter
Referenced by:: ¶ ‣ §10.22(iv), §10.22(iv)
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For the Ferrers function $\mathop{\mathsf{P}\/}\nolimits$ and the associated Legendre function $\mathop{Q\/}\nolimits$ , 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

$A=s(s-a)(s-b)(s-c),$

$s=\tfrac{1}{2}(a+b+c).$

Symbols:: $A$ : area and $s$ : sum
Permalink:: http://dlmf.nist.gov/10.22.E73
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(Thus if $a,b,c$ are the sides of a triangle, then $A^{{\frac{1}{2}}}$ is the area of the triangle.)

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

10.22.74 $\int _{0}^{\infty}\mathop{J_{{\nu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(ct\right)t^{{1-\nu}}dt=\begin{cases}\dfrac{2^{{\nu-1}}A^{{\nu-\frac{1}{2}}}}{\pi^{{\frac{1}{2}}}(abc)^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)},&A>0,\\ 0,&A\leq 0.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\nu$ : complex parameter and $A$ : area
Referenced by:: ¶ ‣ §10.22(iv), §10.22(iv)
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If $|\nu|<\tfrac{1}{2}$ , then

10.22.75 $\int _{0}^{\infty}\mathop{Y_{{\nu}}\/}\nolimits\!\left(at\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(bt\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(ct\right)t^{{1+\nu}}dt=\begin{cases}-\dfrac{(abc)^{\nu}(-A)^{{-\nu-\frac{1}{2}}}}{\pi^{{\frac{1}{2}}}2^{{\nu+1}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\nu\right)},&0<a<|b-c|,\\ 0,&|b-c|<a<b+c,\\ \dfrac{(abc)^{\nu}(-A)^{{-\nu-\frac{1}{2}}}}{\pi^{{\frac{1}{2}}}2^{{\nu+1}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\nu\right)},&a>b+c.\end{cases}$

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, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\nu$ : complex parameter and $A$ : area
Referenced by:: ¶ ‣ §10.22(iv), §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E75
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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’s inversion theorem, Hankel transform, integrals of Bessel and Hankel functions
Referenced by:: Ch.10, ¶ ‣ §3.5(ix)
Permalink:: http://dlmf.nist.gov/10.22.v

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

10.22.76 $g(y)=\int _{0}^{\infty}f(x)\mathop{J_{{\nu}}\/}\nolimits\!\left(xy\right)(xy)^{{\frac{1}{2}}}dx.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $y$ : real variable, $\nu$ : complex parameter, $f(x)$ : function and $g(y)$ : function
Referenced by:: ¶ ‣ §10.74(vii)
Permalink:: http://dlmf.nist.gov/10.22.E76
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Hankel’s inversion theorem is given by

10.22.77 $f(y)=\int _{0}^{\infty}g(x)\mathop{J_{{\nu}}\/}\nolimits\!\left(xy\right)(xy)^{{\frac{1}{2}}}dx.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $y$ : real variable, $\nu$ : complex parameter, $f(x)$ : function and $g(y)$ : function
Referenced by:: §10.22(v)
Permalink:: http://dlmf.nist.gov/10.22.E77
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Sufficient conditions for the validity of (10.22.77) are that $\int _{0}^{\infty}|f(x)|dx<\infty$ when $\nu\geq-\tfrac{1}{2}$ , or that $\int _{0}^{\infty}|f(x)|dx<\infty$ and $\int _{0}^{1}x^{{\nu+\frac{1}{2}}}|f(x)|dx<\infty$ 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) and Frenzen and Wong (1985).

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

§10.22(vi) Compendia

Keywords:: integrals of Bessel and Hankel functions
Permalink:: http://dlmf.nist.gov/10.22.vi

For collections of integrals of the functions $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ , and $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ , 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 (2000, §§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).

§10.22 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

§10.22(iv) Integrals over the Interval

¶ 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 $(0,\infty)$