# §10.22 Integrals

## §10.22(i) Indefinite Integrals

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

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

 10.22.3 $\displaystyle\int e^{iz}z^{\nu}\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(z% \right)\mathrm{d}z$ $\displaystyle=\frac{e^{iz}z^{\nu+1}}{2\nu+1}(\mathop{\mathscr{C}_{\nu}\/}% \nolimits\!\left(z\right)-i\mathop{\mathscr{C}_{\nu+1}\/}\nolimits\!\left(z% \right)),$ $\nu\neq-\tfrac{1}{2}$, $\displaystyle\int e^{iz}z^{-\nu}\mathop{\mathscr{C}_{\nu}\/}\nolimits\!\left(z% \right)\mathrm{d}z$ $\displaystyle=\frac{e^{iz}z^{-\nu+1}}{1-2\nu}(\mathop{\mathscr{C}_{\nu}\/}% \nolimits\!\left(z\right)+i\mathop{\mathscr{C}_{\nu-1}\/}\nolimits\!\left(z% \right)),$ $\nu\neq\tfrac{1}{2}$.

### Products

 10.22.4 $\int z\mathop{\mathscr{C}_{\mu}\/}\nolimits\!\left(az\right)\mathscr{D}_{\mu}(% bz)\mathrm{d}z=\frac{z\left(a\mathop{\mathscr{C}_{\mu+1}\/}\nolimits\!\left(az% \right)\mathscr{D}_{\mu}(bz)-b\mathop{\mathscr{C}_{\mu}\/}\nolimits\!\left(az% \right)\mathscr{D}_{\mu+1}(bz)\right)}{a^{2}-b^{2}},$ $a^{2}\neq b^{2}$,
 10.22.5 $\displaystyle\int z\mathop{\mathscr{C}_{\mu}\/}\nolimits\!\left(az\right)% \mathscr{D}_{\mu}(az)\mathrm{d}z$ $\displaystyle=\tfrac{1}{4}z^{2}\left(2\mathop{\mathscr{C}_{\mu}\/}\nolimits\!% \left(az\right)\mathscr{D}_{\mu}(az)-\mathop{\mathscr{C}_{\mu-1}\/}\nolimits\!% \left(az\right)\mathscr{D}_{\mu+1}(az)-\mathop{\mathscr{C}_{\mu+1}\/}\nolimits% \!\left(az\right)\mathscr{D}_{\mu-1}(az)\right),$ 10.22.6 $\displaystyle\int\mathop{\mathscr{C}_{\mu}\/}\nolimits\!\left(az\right)% \mathscr{D}_{\nu}(az)\frac{\mathrm{d}z}{z}$ $\displaystyle=-\frac{az(\mathop{\mathscr{C}_{\mu+1}\/}\nolimits\!\left(az% \right)\mathscr{D}_{\nu}(az)-\mathop{\mathscr{C}_{\mu}\/}\nolimits\!\left(az% \right)\mathscr{D}_{\nu+1}(az))}{\mu^{2}-\nu^{2}}+\frac{\mathop{\mathscr{C}_{% \mu}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)}{\mu+\nu},$ $\mu^{2}\neq\nu^{2}$,
 10.22.7 $\displaystyle\int z^{\mu+\nu+1}\mathop{\mathscr{C}_{\mu}\/}\nolimits\!\left(az% \right)\mathscr{D}_{\nu}(az)\mathrm{d}z$ $\displaystyle=\frac{z^{\mu+\nu+2}}{2(\mu+\nu+1)}\*\left(\mathop{\mathscr{C}_{% \mu}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)+\mathop{\mathscr{C}_{% \mu+1}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu+1}(az)\right),$ $\mu+\nu\neq-1$, $\displaystyle\int z^{-\mu-\nu+1}\mathop{\mathscr{C}_{\mu}\/}\nolimits\!\left(% az\right)\mathscr{D}_{\nu}(az)\mathrm{d}z$ $\displaystyle=\frac{z^{-\mu-\nu+2}}{2(1-\mu-\nu)}\*\left(\mathop{\mathscr{C}_{% \mu}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu}(az)+\mathop{\mathscr{C}_{% \mu-1}\/}\nolimits\!\left(az\right)\mathscr{D}_{\nu-1}(az)\right),$ $\mu+\nu\neq 1$.

## §10.22(ii) Integrals over Finite Intervals

Throughout this subsection $x>0$.

 10.22.8 $\int_{0}^{x}\mathop{J_{\nu}\/}\nolimits\!\left(t\right)\mathrm{d}t=2\sum_{k=0}% ^{\infty}\mathop{J_{\nu+2k+1}\/}\nolimits\!\left(x\right),$ $\Re{\nu}>-1$.
 10.22.9 $\int_{0}^{x}\mathop{J_{2n}\/}\nolimits\!\left(t\right)\mathrm{d}t=\int_{0}^{x}% \mathop{J_{0}\/}\nolimits\!\left(t\right)\mathrm{d}t-2\sum_{k=0}^{n-1}\mathop{% J_{2k+1}\/}\nolimits\!\left(x\right),\quad\int_{0}^{x}\mathop{J_{2n+1}\/}% \nolimits\!\left(t\right)\mathrm{d}t=1-\mathop{J_{0}\/}\nolimits\!\left(x% \right)-2\sum_{k=1}^{n}\mathop{J_{2k}\/}\nolimits\!\left(x\right),$ $n=0,1,\dots$.
 10.22.10 $\int_{0}^{x}t^{\mu}\mathop{J_{\nu}\/}\nolimits\!\left(t\right)\mathrm{d}t=x^{% \mu}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+% \frac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{% 2}\mu+\frac{1}{2}\right)}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\mathop{\Gamma\/% }\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k\right)}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}% \mathop{J_{\nu+2k+1}\/}\nolimits\!\left(x\right),$ $\Re{(\mu+\nu+1)}>0$.
 10.22.11 $\displaystyle\int_{0}^{x}\frac{1-\mathop{J_{0}\/}\nolimits\!\left(t\right)}{t}% \mathrm{d}t$ $\displaystyle=\frac{1}{2}\sum_{k=1}^{\infty}\frac{\mathop{\psi\/}\nolimits\!% \left(k+1\right)-\mathop{\psi\/}\nolimits\!\left(1\right)}{k!}(\tfrac{1}{2}x)^% {k}\mathop{J_{k}\/}\nolimits\!\left(x\right),$ 10.22.12 $\displaystyle x\int_{0}^{x}\frac{1-\mathop{J_{0}\/}\nolimits\!\left(t\right)}{% t}\mathrm{d}t$ $\displaystyle=2\sum_{k=0}^{\infty}(2k+3)(\mathop{\psi\/}\nolimits\!\left(k+2% \right)-\mathop{\psi\/}\nolimits\!\left(1\right))\mathop{J_{2k+3}\/}\nolimits% \!\left(x\right)$ $\displaystyle=x-2\!\mathop{J_{1}\/}\nolimits\!\left(x\right)+2\sum_{k=0}^{% \infty}(2k+5)\*(\mathop{\psi\/}\nolimits\!\left(k+3\right)-\mathop{\psi\/}% \nolimits\!\left(1\right)-1)\mathop{J_{2k+5}\/}\nolimits\!\left(x\right),$

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

### Trigonometric Arguments

 10.22.13 $\displaystyle\int_{0}^{\frac{1}{2}\pi}\mathop{J_{2\nu}\/}\nolimits\!\left(2z% \mathop{\cos\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits\!\left(2\mu% \theta\right)\mathrm{d}\theta$ $\displaystyle=\tfrac{1}{2}\pi\mathop{J_{\nu+\mu}\/}\nolimits\!\left(z\right)% \mathop{J_{\nu-\mu}\/}\nolimits\!\left(z\right),$ $\Re{\nu}>-\tfrac{1}{2}$, Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: 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) 10.22.14 $\displaystyle\int_{0}^{\pi}\mathop{J_{2\nu}\/}\nolimits\!\left(2z\mathop{\sin% \/}\nolimits\theta\right)\mathop{\cos\/}\nolimits\!\left(2\mu\theta\right)% \mathrm{d}\theta$ $\displaystyle=\pi\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)\mathop{J_{\nu+% \mu}\/}\nolimits\!\left(z\right)\mathop{J_{\nu-\mu}\/}\nolimits\!\left(z\right),$ $\Re{\nu}>-\tfrac{1}{2}$, Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\Re{}$: real part, $\mathop{\sin\/}\nolimits\NVar{z}$: 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) 10.22.15 $\displaystyle\int_{0}^{\pi}\mathop{J_{2\nu}\/}\nolimits\!\left(2z\mathop{\sin% \/}\nolimits\theta\right)\mathop{\sin\/}\nolimits(2\mu\theta)\mathrm{d}\theta$ $\displaystyle=\pi\mathop{\sin\/}\nolimits(\mu\pi)\mathop{J_{\nu+\mu}\/}% \nolimits\!\left(z\right)\mathop{J_{\nu-\mu}\/}\nolimits\!\left(z\right),$ $\Re{\nu}>-1$. 10.22.16 $\displaystyle\int_{0}^{\frac{1}{2}\pi}\mathop{J_{0}\/}\nolimits\!\left(2z% \mathop{\sin\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits(2n\theta)\mathrm% {d}\theta$ $\displaystyle=\tfrac{1}{2}\pi{\mathop{J_{n}\/}\nolimits^{2}}\!\left(z\right),$ $n=0,1,2,\ldots$.
 10.22.17 $\int_{0}^{\frac{1}{2}\pi}\mathop{Y_{2\nu}\/}\nolimits\!\left(2z\mathop{\cos\/}% \nolimits\theta\right)\mathop{\cos\/}\nolimits(2\mu\theta)\mathrm{d}\theta=% \tfrac{1}{2}\pi\mathop{\cot\/}\nolimits(2\nu\pi)\mathop{J_{\nu+\mu}\/}% \nolimits\!\left(z\right)\mathop{J_{\nu-\mu}\/}\nolimits\!\left(z\right)-% \tfrac{1}{2}\pi\mathop{\csc\/}\nolimits(2\nu\pi)\mathop{J_{\mu-\nu}\/}% \nolimits\!\left(z\right)\mathop{J_{-\mu-\nu}\/}\nolimits\!\left(z\right),$ $-\tfrac{1}{2}<\Re{\nu}<\tfrac{1}{2}$,
 10.22.18 $\int_{0}^{\frac{1}{2}\pi}\mathop{Y_{0}\/}\nolimits\!\left(2z\mathop{\sin\/}% \nolimits\theta\right)\mathop{\cos\/}\nolimits\!\left(2n\theta\right)\mathrm{d% }\theta=\tfrac{1}{2}\pi\mathop{J_{n}\/}\nolimits\!\left(z\right)\mathop{Y_{n}% \/}\nolimits\!\left(z\right),$ $n=0,1,2,\dots$.
 10.22.19 $\int_{0}^{\frac{1}{2}\pi}\mathop{J_{\mu}\/}\nolimits\!\left(z\mathop{\sin\/}% \nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{\mu+1}(\mathop{\cos\/}% \nolimits\theta)^{2\nu+1}\mathrm{d}\theta=2^{\nu}\mathop{\Gamma\/}\nolimits\!% \left(\nu+1\right)z^{-\nu-1}\mathop{J_{\mu+\nu+1}\/}\nolimits\!\left(z\right),$ $\Re{\mu}>-1$, $\Re{\nu}>-1$,
 10.22.20 $\displaystyle\int_{0}^{\frac{1}{2}\pi}\mathop{J_{\mu}\/}\nolimits\!\left(z% \mathop{\sin\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{\mu}(% \mathop{\cos\/}\nolimits\theta)^{2\mu}\mathrm{d}\theta$ $\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\mathop{\Gamma\/}\nolimits\!% \left(\mu+\tfrac{1}{2}\right){\mathop{J_{\mu}\/}\nolimits^{2}}\!\left(\tfrac{1% }{2}z\right),$ $\Re{\mu}>-\tfrac{1}{2}$, 10.22.21 $\displaystyle\int_{0}^{\frac{1}{2}\pi}\mathop{Y_{\mu}\/}\nolimits\!\left(z% \mathop{\sin\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{\mu}(% \mathop{\cos\/}\nolimits\theta)^{2\mu}\mathrm{d}\theta$ $\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\mathop{\Gamma\/}\nolimits\!% \left(\mu+\tfrac{1}{2}\right)\mathop{J_{\mu}\/}\nolimits\!\left(\tfrac{1}{2}z% \right)\mathop{Y_{\mu}\/}\nolimits\!\left(\tfrac{1}{2}z\right),$ $\Re{\mu}>-\tfrac{1}{2}$.
 10.22.22 $\int_{0}^{\frac{1}{2}\pi}\mathop{J_{\mu}\/}\nolimits\!\left(z{\mathop{\sin\/}% \nolimits^{2}}\theta\right)\mathop{J_{\nu}\/}\nolimits\!\left(z{\mathop{\cos\/% }\nolimits^{2}}\theta\right)(\mathop{\sin\/}\nolimits\theta)^{2\mu+1}(\mathop{% \cos\/}\nolimits\theta)^{2\nu+1}\mathrm{d}\theta=\frac{\mathop{\Gamma\/}% \nolimits\!\left(\mu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+% \tfrac{1}{2}\right)\mathop{J_{\mu+\nu+\frac{1}{2}}\/}\nolimits\!\left(z\right)% }{(8\pi z)^{\frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu+1\right)},$ $\Re{\mu}>-\tfrac{1}{2},\Re{\nu}>-\tfrac{1}{2}$.
 10.22.23 $\displaystyle\int_{0}^{\frac{1}{2}\pi}\mathop{J_{\mu}\/}\nolimits\!\left(z{% \mathop{\sin\/}\nolimits^{2}}\theta\right)\mathop{J_{\nu}\/}\nolimits\!\left(z% {\mathop{\cos\/}\nolimits^{2}}\theta\right)(\mathop{\sin\/}\nolimits\theta)^{2% \alpha-1}\mathop{\sec\/}\nolimits\theta\mathrm{d}\theta$ $\displaystyle=\frac{(\mu+\nu+\alpha)\mathop{\Gamma\/}\nolimits\!\left(\mu+% \alpha\right)2^{\alpha-1}}{\nu\mathop{\Gamma\/}\nolimits\!\left(\mu+1\right)z^% {\alpha}}\mathop{J_{\mu+\nu+\alpha}\/}\nolimits\!\left(z\right),$ $\Re{(\mu+\alpha)}>0$, $\Re{\nu}>0$. 10.22.24 $\displaystyle\int_{0}^{\frac{1}{2}\pi}\mathop{J_{\mu}\/}\nolimits\!\left(z{% \mathop{\sin\/}\nolimits^{2}}\theta\right)\mathop{J_{\nu}\/}\nolimits\!\left(z% {\mathop{\cos\/}\nolimits^{2}}\theta\right)\mathop{\cot\/}\nolimits\theta% \mathrm{d}\theta$ $\displaystyle=\tfrac{1}{2}\mu^{-1}\mathop{J_{\mu+\nu}\/}\nolimits\!\left(z% \right),$ $\Re{\mu}>0,\Re{\nu}>-1$. 10.22.25 $\displaystyle\int_{0}^{\frac{1}{2}\pi}\mathop{J_{\mu}\/}\nolimits\!\left(z% \mathop{\sin\/}\nolimits\theta\right)\mathop{I_{\nu}\/}\nolimits\!\left(z% \mathop{\cos\/}\nolimits\theta\right)(\mathop{\tan\/}\nolimits\theta)^{\mu+1}% \mathrm{d}\theta$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\nu-\tfrac{1% }{2}\mu\right)(\tfrac{1}{2}z)^{\mu}}{2\!\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}\mathop{J_{\nu}\/}\nolimits\!\left(z% \right),$ $\Re{\nu}>\Re{\mu}>-1$.

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

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

### Products

 10.22.27 $\displaystyle\int_{0}^{x}t{\mathop{J_{\nu-1}\/}\nolimits^{2}}\!\left(t\right)% \mathrm{d}t$ $\displaystyle=2\sum_{k=0}^{\infty}(\nu+2k){\mathop{J_{\nu+2k}\/}\nolimits^{2}}% \!\left(x\right),$ $\Re{\nu}>0$, 10.22.28 $\displaystyle\int_{0}^{x}t\left({\mathop{J_{\nu-1}\/}\nolimits^{2}}\!\left(t% \right)-{\mathop{J_{\nu+1}\/}\nolimits^{2}}\!\left(t\right)\right)\mathrm{d}t$ $\displaystyle=2\nu{\mathop{J_{\nu}\/}\nolimits^{2}}\!\left(x\right),$ $\Re{\nu}>0$, 10.22.29 $\displaystyle\int_{0}^{x}t{\mathop{J_{0}\/}\nolimits^{2}}\!\left(t\right)% \mathrm{d}t$ $\displaystyle=\tfrac{1}{2}x^{2}\left({\mathop{J_{0}\/}\nolimits^{2}}\!\left(x% \right)+{\mathop{J_{1}\/}\nolimits^{2}}\!\left(x\right)\right).$ Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 10.22.30 $\int_{0}^{x}\mathop{J_{n}\/}\nolimits\!\left(t\right)\mathop{J_{n+1}\/}% \nolimits\!\left(t\right)\mathrm{d}t=\tfrac{1}{2}\left(1-{\mathop{J_{0}\/}% \nolimits^{2}}\!\left(x\right)\right)-\sum_{k=1}^{n}{\mathop{J_{k}\/}\nolimits% ^{2}}\!\left(x\right)=\sum_{k=n+1}^{\infty}{\mathop{J_{k}\/}\nolimits^{2}}\!% \left(x\right),$ $n=0,1,2,\ldots$.

### Convolutions

 10.22.31 $\int_{0}^{x}\mathop{J_{\mu}\/}\nolimits\!\left(t\right)\mathop{J_{\nu}\/}% \nolimits\!\left(x-t\right)\mathrm{d}t=2\sum_{k=0}^{\infty}(-1)^{k}\mathop{J_{% \mu+\nu+2k+1}\/}\nolimits\!\left(x\right),$ $\Re{\mu}>-1,\Re{\nu}>-1$.
 10.22.32 $\displaystyle\int_{0}^{x}\mathop{J_{\nu}\/}\nolimits\!\left(t\right)\mathop{J_% {1-\nu}\/}\nolimits\!\left(x-t\right)\mathrm{d}t$ $\displaystyle=\mathop{J_{0}\/}\nolimits\!\left(x\right)-\mathop{\cos\/}% \nolimits x,$ $-1<\Re{\nu}<2$. 10.22.33 $\displaystyle\int_{0}^{x}\mathop{J_{\nu}\/}\nolimits\!\left(t\right)\mathop{J_% {-\nu}\/}\nolimits\!\left(x-t\right)\mathrm{d}t$ $\displaystyle=\mathop{\sin\/}\nolimits x,$ $|\Re{\nu}|<1$.
 10.22.34 $\int_{0}^{x}t^{-1}\mathop{J_{\mu}\/}\nolimits\!\left(t\right)\mathop{J_{\nu}\/% }\nolimits\!\left(x-t\right)\mathrm{d}t=\frac{\mathop{J_{\mu+\nu}\/}\nolimits% \!\left(x\right)}{\mu},$ $\Re{\mu}>0,\Re{\nu}>-1$.
 10.22.35 $\int_{0}^{x}\frac{\mathop{J_{\mu}\/}\nolimits\!\left(t\right)\mathop{J_{\nu}\/% }\nolimits\!\left(x-t\right)\mathrm{d}t}{t(x-t)}=\frac{(\mu+\nu)\mathop{J_{\mu% +\nu}\/}\nolimits\!\left(x\right)}{\mu\nu x},$ $\Re{\mu}>0,\Re{\nu}>0$.

### Fractional Integral

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

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

If $\nu>-1$, then

 10.22.37 $\int_{0}^{1}t\mathop{J_{\nu}\/}\nolimits\!\left(j_{\nu,\ell}t\right)\mathop{J_% {\nu}\/}\nolimits\!\left(j_{\nu,m}t\right)\mathrm{d}t=\tfrac{1}{2}\delta_{\ell% ,m}\left(\mathop{J_{\nu}\/}\nolimits'\!\left(j_{\nu,\ell}\right)\right)^{2},$

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 $\int_{0}^{1}t\mathop{J_{\nu}\/}\nolimits\!\left(\alpha_{\ell}t\right)\mathop{J% _{\nu}\/}\nolimits\!\left(\alpha_{m}t\right)\mathrm{d}t=\delta_{\ell,m}\left(% \frac{a^{2}}{b^{2}}+\alpha_{\ell}^{2}-\nu^{2}\right)\frac{(\mathop{J_{\nu}\/}% \nolimits\!\left(\alpha_{\ell}\right))^{2}}{2\alpha_{\ell}^{2}},$

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

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

When $x>0$

 10.22.39 $\int_{x}^{\infty}\frac{\mathop{J_{0}\/}\nolimits\!\left(t\right)}{t}\mathrm{d}% t+\gamma+\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)=\int_{0}^{x}\frac% {1-\mathop{J_{0}\/}\nolimits\!\left(t\right)}{t}\mathrm{d}t=\sum_{k=1}^{\infty% }(-1)^{k-1}\frac{(\frac{1}{2}x)^{2k}}{2k(k!)^{2}},$
 10.22.40 $\int_{x}^{\infty}\frac{\mathop{Y_{0}\/}\nolimits\!\left(t\right)}{t}\mathrm{d}% t=-\frac{1}{\pi}\left(\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}x\right)+% \gamma\right)^{2}+\frac{\pi}{6}+\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\*% \left(\mathop{\psi\/}\nolimits\!\left(k+1\right)+\frac{1}{2k}-\mathop{\ln\/}% \nolimits\!\left(\tfrac{1}{2}x\right)\right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)% ^{2}},$

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

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

 10.22.41 $\displaystyle\int_{0}^{\infty}\mathop{J_{\nu}\/}\nolimits\!\left(t\right)% \mathrm{d}t$ $\displaystyle=1,$ $\Re{\nu}>-1$, 10.22.42 $\displaystyle\int_{0}^{\infty}\mathop{Y_{\nu}\/}\nolimits\!\left(t\right)% \mathrm{d}t$ $\displaystyle=-\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\nu\pi\right),$ $|\Re{\nu}|<1$.
 10.22.43 $\displaystyle\int_{0}^{\infty}t^{\mu}\mathop{J_{\nu}\/}\nolimits\!\left(t% \right)\mathrm{d}t$ $\displaystyle=2^{\mu}\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\nu+% \tfrac{1}{2}\mu+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{% 1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},$ $\Re{(\mu+\nu)}>-1$, $\Re{\mu}<\tfrac{1}{2}$, 10.22.44 $\displaystyle\int_{0}^{\infty}t^{\mu}\mathop{Y_{\nu}\/}\nolimits\!\left(t% \right)\mathrm{d}t$ $\displaystyle=\frac{2^{\mu}}{\pi}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2% }\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\mathop{\sin\/}\nolimits\!% \left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi,$ $\Re{(\mu\pm\nu)}>-1$, $\Re{\mu}<\tfrac{1}{2}$.
 10.22.45 $\int_{0}^{\infty}\frac{1-\mathop{J_{0}\/}\nolimits\!\left(t\right)}{t^{\mu}}% \mathrm{d}t=-\frac{\pi\mathop{\sec\/}\nolimits\!\left(\frac{1}{2}\mu\pi\right)% }{2^{\mu}{\mathop{\Gamma\/}\nolimits^{2}}\!\left(\frac{1}{2}\mu+\frac{1}{2}% \right)},$ $1<\Re{\mu}<3$.
 10.22.46 $\int_{0}^{\infty}\frac{t^{\nu+1}\mathop{J_{\nu}\/}\nolimits\!\left(at\right)}{% (t^{2}+b^{2})^{\mu+1}}\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\mathop{% \Gamma\/}\nolimits\!\left(\mu+1\right)}\mathop{K_{\nu-\mu}\/}\nolimits\!\left(% ab\right),$ $a>0$, $\Re{b}>0$, $-1<\Re{\nu}<2\Re{\mu}+\tfrac{3}{2}$.
 10.22.47 $\int_{0}^{\infty}\frac{t^{\nu}\mathop{Y_{\nu}\/}\nolimits\!\left(at\right)}{t^% {2}+b^{2}}\mathrm{d}t=-b^{\nu-1}\mathop{K_{\nu}\/}\nolimits\!\left(ab\right),$ $a>0,\Re{b}>0,-\tfrac{1}{2}<\Re{\nu}<\tfrac{5}{2}$. Symbols: $\mathop{Y_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the second kind, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{K_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\Re{}$: 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 See also: Annotations for 10.22(iv)

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

 10.22.48 $\int_{0}^{\infty}\mathop{J_{\mu}\/}\nolimits\!\left(x\mathop{\cosh\/}\nolimits% \phi\right)(\mathop{\cosh\/}\nolimits\phi)^{1-\mu}(\mathop{\sinh\/}\nolimits% \phi)^{2\nu+1}\mathrm{d}\phi=2^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu+1% \right)x^{-\nu-1}\mathop{J_{\mu-\nu-1}\/}\nolimits\!\left(x\right),$ $x>0,\Re{\nu}>-1,\Re{\mu}>2\Re{\nu}+\tfrac{1}{2}$.
 10.22.49 $\int_{0}^{\infty}t^{\mu-1}e^{-at}\mathop{J_{\nu}\/}\nolimits\!\left(bt\right)% \mathrm{d}t=\frac{(\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\mathop{\Gamma\/}% \nolimits\!\left(\mu+\nu\right)\*\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{% \mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),$ $\Re{(\mu+\nu)}>0,\Re{(a\pm ib)}>0$,
 10.22.50 $\int_{0}^{\infty}t^{\mu-1}e^{-at}\mathop{Y_{\nu}\/}\nolimits\!\left(bt\right)% \mathrm{d}t=\mathop{\cot\/}\nolimits(\nu\pi)\frac{(\tfrac{1}{2}b)^{\nu}\mathop% {\Gamma\/}\nolimits\!\left(\mu+\nu\right)}{(a^{2}+b^{2})^{\frac{1}{2}(\mu+\nu)% }}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{\mu+\nu}{2},\frac{1-\mu+\nu}{2% };\nu+1;\frac{b^{2}}{a^{2}+b^{2}}\right)-\mathop{\csc\/}\nolimits(\nu\pi)\frac% {(\tfrac{1}{2}b)^{-\nu}\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)}{(a^{2% }+b^{2})^{\frac{1}{2}(\mu-\nu)}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{% \mu-\nu}{2},\frac{1-\mu-\nu}{2};1-\nu;\frac{b^{2}}{a^{2}+b^{2}}\right),$ $\Re{\mu}>|\Re{\nu}|,\Re{(a\pm ib)}>0$.

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

 10.22.51 $\displaystyle\int_{0}^{\infty}\mathop{J_{\nu}\/}\nolimits\!\left(bt\right)% \mathop{\exp\/}\nolimits\!\left(-p^{2}t^{2}\right)t^{\nu+1}\mathrm{d}t$ $\displaystyle=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\mathop{\exp\/}\nolimits\!\left(% -\frac{b^{2}}{4p^{2}}\right),$ $\Re{\nu}>-1$, $\Re{(p^{2})}>0$, 10.22.52 $\displaystyle\int_{0}^{\infty}\mathop{J_{\nu}\/}\nolimits\!\left(bt\right)% \mathop{\exp\/}\nolimits(-p^{2}t^{2})\mathrm{d}t$ $\displaystyle=\frac{\sqrt{\pi}}{2p}\mathop{\exp\/}\nolimits\left(-\frac{b^{2}}% {8p^{2}}\right)\mathop{I_{\ifrac{\nu}{2}}\/}\nolimits\left(\frac{b^{2}}{8p^{2}% }\right),$ $\Re{\nu}>-1,\Re{(p^{2})}>0$,
 10.22.53 $\int_{0}^{\infty}\mathop{Y_{2\nu}\/}\nolimits\!\left(bt\right)\mathop{\exp\/}% \nolimits\!\left(-p^{2}t^{2}\right)\mathrm{d}t=-\frac{\sqrt{\pi}}{2p}\mathop{% \exp\/}\nolimits\!\left(-\frac{b^{2}}{8p^{2}}\right)\left(\mathop{I_{\nu}\/}% \nolimits\left(\frac{b^{2}}{8p^{2}}\right)\mathop{\tan\/}\nolimits\!\left(\nu% \pi\right)+\frac{1}{\pi}\mathop{K_{\nu}\/}\nolimits\left(\frac{b^{2}}{8p^{2}}% \right)\mathop{\sec\/}\nolimits\!\left(\nu\pi\right)\right),$ $|\Re{\nu}|<\tfrac{1}{2}$, $\Re{(p^{2})}>0$.

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

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

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

### Orthogonality

 10.22.55 $\int_{0}^{\infty}t^{-1}\mathop{J_{\nu+2\ell+1}\/}\nolimits\!\left(t\right)% \mathop{J_{\nu+2m+1}\/}\nolimits\!\left(t\right)\mathrm{d}t=\frac{\delta_{\ell% ,m}}{2(2\ell+\nu+1)},$ $\nu+\ell+m>-1$.

### Weber–Schafheitlin Discontinuous Integrals, including Special Cases

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

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

 10.22.57 $\displaystyle\int_{0}^{\infty}\frac{\mathop{J_{\mu}\/}\nolimits\!\left(at% \right)\mathop{J_{\nu}\/}\nolimits\!\left(at\right)}{t^{\lambda}}\mathrm{d}t$ $\displaystyle=\frac{(\frac{1}{2}a)^{\lambda-1}\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}\mu+\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\right)% \mathop{\Gamma\/}\nolimits\!\left(\lambda\right)}{2\mathop{\Gamma\/}\nolimits% \!\left(\frac{1}{2}\lambda+\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\lambda+\frac{1}{2}\mu-\frac{1}{2% }\nu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\lambda+% \frac{1}{2}\mu+\frac{1}{2}\nu+\frac{1}{2}\right)},$ $\Re{(\mu+\nu+1)}>\Re{\lambda}>0$. 10.22.58 $\displaystyle\int_{0}^{\infty}\frac{\mathop{J_{\nu}\/}\nolimits\!\left(at% \right)\mathop{J_{\nu}\/}\nolimits\!\left(bt\right)}{t^{\lambda}}\mathrm{d}t$ $\displaystyle=\frac{(ab)^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu-\frac{1}{2% }\lambda+\frac{1}{2}\right)}{2^{\lambda}(a^{2}+b^{2})^{\nu-\frac{1}{2}\lambda+% \frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\lambda+\frac{1}{2}% \right)}\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{2\nu+1-\lambda}{4},\frac{2% \nu+3-\lambda}{4};\nu+1;\frac{4a^{2}b^{2}}{(a^{2}+b^{2})^{2}}\right),$ $a\neq b$, $\Re{(2\nu+1)}>\Re{\lambda}>-1$.

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

 10.22.59 $\int_{0}^{\infty}e^{ibt}\mathop{J_{\mu}\/}\nolimits\!\left(at\right)\mathrm{d}% t=\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
 10.22.60 $\int_{0}^{\infty}e^{ibt}\mathop{Y_{0}\/}\nolimits\!\left(at\right)\mathrm{d}t=% \begin{cases}(2i/\pi)(a^{2}-b^{2})^{-\frac{1}{2}}\mathop{\mathrm{arcsin}\/}% \nolimits\!\left(b/a\right),&0\leq b

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

 10.22.61 $\int_{0}^{\infty}t^{-1}e^{ibt}\mathop{J_{\mu}\/}\nolimits\!\left(at\right)% \mathrm{d}t=\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

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

 10.22.62 $\int_{0}^{\infty}t^{\mu-\nu+1}\mathop{J_{\mu}\/}\nolimits\!\left(at\right)% \mathop{J_{\nu}\/}\nolimits\!\left(bt\right)\mathrm{d}t=\begin{cases}0,&0

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

 10.22.63 $\int_{0}^{\infty}\mathop{J_{\mu}\/}\nolimits\!\left(at\right)\mathop{J_{\mu-1}% \/}\nolimits\!\left(bt\right)\mathrm{d}t=\begin{cases}b^{\mu-1}a^{-\mu},&00),\\ 0,&0 Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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

 10.22.64 $\int_{0}^{\infty}\mathop{J_{\mu+2n+1}\/}\nolimits\!\left(at\right)\mathop{J_{% \mu}\/}\nolimits\!\left(bt\right)\mathrm{d}t=\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),&00),\\ 0,&0
 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{\mathrm{d}t}{t}=\begin{cases}0,&0\leq b Symbols: $\mathop{J_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\mathop{\ln\/}\nolimits\NVar{z}$: 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)

### Other Double Products

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

 10.22.66 $\displaystyle\int_{0}^{\infty}e^{-at}\mathop{J_{\nu}\/}\nolimits\!\left(bt% \right)\mathop{J_{\nu}\/}\nolimits\!\left(ct\right)\mathrm{d}t$ $\displaystyle=\frac{1}{\pi(bc)^{\frac{1}{2}}}\*\mathop{Q_{\nu-\frac{1}{2}}\/}% \nolimits\left(\frac{a^{2}+b^{2}+c^{2}}{2bc}\right),$ $\Re{\nu}>-\tfrac{1}{2}$. 10.22.67 $\displaystyle\int_{0}^{\infty}t\mathop{\exp\/}\nolimits(-p^{2}t^{2})\mathop{J_% {\nu}\/}\nolimits\!\left(at\right)\mathop{J_{\nu}\/}\nolimits\!\left(bt\right)% \mathrm{d}t$ $\displaystyle=\frac{1}{2p^{2}}\mathop{\exp\/}\nolimits\left(-\frac{a^{2}+b^{2}% }{4p^{2}}\right)\mathop{I_{\nu}\/}\nolimits\left(\frac{ab}{2p^{2}}\right),$ $\Re{\nu}>-1,\Re{(p^{2})}>0$. 10.22.68 $\displaystyle\int_{0}^{\infty}t\mathop{\exp\/}\nolimits(-p^{2}t^{2})\mathop{J_% {0}\/}\nolimits\!\left(at\right)\mathop{Y_{0}\/}\nolimits\!\left(at\right)% \mathrm{d}t$ $\displaystyle=-\frac{1}{2\pi p^{2}}\mathop{\exp\/}\nolimits\left(-\frac{a^{2}}% {2p^{2}}\right)\mathop{K_{0}\/}\nolimits\left(\frac{a^{2}}{2p^{2}}\right),$ $\Re{(p^{2})}>0$.

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 $\displaystyle\int_{0}^{\infty}\mathop{J_{\nu}\/}\nolimits\!\left(at\right)% \mathop{J_{\nu}\/}\nolimits\!\left(bt\right)\frac{t\mathrm{d}t}{t^{2}-z^{2}}$ $\displaystyle=\left\{\begin{array}[]{ll}\frac{1}{2}\pi i\mathop{J_{\nu}\/}% \nolimits\!\left(bz\right)\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(az\right)% ,&a>b\\ \frac{1}{2}\pi i\mathop{J_{\nu}\/}\nolimits\!\left(az\right)\mathop{{H^{(1)}_{% \nu}}\/}\nolimits\!\left(bz\right),&b>a\end{array}\right\},$ $\Re{\nu}>-1,\Im{z}>0$. 10.22.70 $\displaystyle\int_{0}^{\infty}\mathop{Y_{\nu}\/}\nolimits\!\left(at\right)% \mathop{J_{\nu+1}\/}\nolimits\!\left(bt\right)\frac{t\mathrm{d}t}{t^{2}-z^{2}}$ $\displaystyle=\frac{1}{2}\pi\mathop{J_{\nu+1}\/}\nolimits\!\left(bz\right)% \mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(az\right),$ $a\geq b>0$, $\Re{\nu}>-\tfrac{3}{2},\Im{z}>0$.

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 $\Re{\nu}>-\frac{3}{2}$ 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

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

 10.22.71 $\displaystyle\int_{0}^{\infty}\mathop{J_{\mu}\/}\nolimits\!\left(at\right)% \mathop{J_{\nu}\/}\nolimits\!\left(bt\right)\mathop{J_{\nu}\/}\nolimits\!\left% (ct\right)t^{1-\mu}\mathrm{d}t$ $\displaystyle=\frac{(bc)^{\mu-1}(\mathop{\sin\/}\nolimits\phi)^{\mu-\frac{1}{2% }}}{(2\pi)^{\frac{1}{2}}a^{\mu}}\mathop{\mathsf{P}^{\frac{1}{2}-\mu}_{\nu-% \frac{1}{2}}\/}\nolimits(\mathop{\cos\/}\nolimits\phi),$ $\Re{\mu}>-\tfrac{1}{2},\Re{\nu}>-1,|b-c|. 10.22.72 $\displaystyle\int_{0}^{\infty}\mathop{J_{\mu}\/}\nolimits\!\left(at\right)% \mathop{J_{\nu}\/}\nolimits\!\left(bt\right)\mathop{J_{\nu}\/}\nolimits\!\left% (ct\right)t^{1-\mu}\mathrm{d}t$ $\displaystyle=\frac{(bc)^{\mu-1}\mathop{\cos\/}\nolimits(\nu\pi)(\mathop{\sinh% \/}\nolimits\chi)^{\mu-\frac{1}{2}}}{(\frac{1}{2}\pi^{3})^{\frac{1}{2}}a^{\mu}% }\mathop{Q^{\frac{1}{2}-\mu}_{\nu-\frac{1}{2}}\/}\nolimits(\mathop{\cosh\/}% \nolimits\chi),$ $\Re{\mu}>-\tfrac{1}{2},\Re{\nu}>-1,a>b+c,\mathop{\cosh\/}\nolimits\chi=(a^{2}-% b^{2}-c^{2})/(2bc)$.

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

(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 $\displaystyle\int_{0}^{\infty}\mathop{J_{\nu}\/}\nolimits\!\left(at\right)% \mathop{J_{\nu}\/}\nolimits\!\left(bt\right)\mathop{J_{\nu}\/}\nolimits\!\left% (ct\right)t^{1-\nu}\mathrm{d}t$ $\displaystyle=\begin{cases}\dfrac{2^{\nu-1}A^{\nu-\frac{1}{2}}}{\pi^{\frac{1}{% 2}}(abc)^{\nu}\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)},&A>0,% \\ 0,&A\leq 0.\end{cases}$ If $|\nu|<\tfrac{1}{2}$, then 10.22.75 $\displaystyle\int_{0}^{\infty}\mathop{Y_{\nu}\/}\nolimits\!\left(at\right)% \mathop{J_{\nu}\/}\nolimits\!\left(bt\right)\mathop{J_{\nu}\/}\nolimits\!\left% (ct\right)t^{1+\nu}\mathrm{d}t$ $\displaystyle=\begin{cases}-\dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{% \frac{1}{2}}2^{\nu+1}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\nu\right)}% ,&0b+c.\end{cases}$

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

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

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

Hankel’s inversion theorem is given by

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

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).

## §10.22(vi) Compendia

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).