# §10.22 Integrals

## §10.22(i) Indefinite Integrals

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

 10.22.1 $\displaystyle\int z^{\nu+1}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=z^{\nu+1}\mathscr{C}_{\nu+1}\left(z\right),$ $\displaystyle\int z^{-\nu+1}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=-z^{-\nu+1}\mathscr{C}_{\nu-1}\left(z\right).$ ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $z$: complex variable and $\nu$: complex parameter A&S Ref: 11.3.20, 11.3.21 (modified) Referenced by: §10.22(i) Permalink: http://dlmf.nist.gov/10.22.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.22(i), 10.22 and 10
 10.22.2 $\int z^{\nu}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z=\pi^{\frac{1}{2}}2^{\nu% -1}\Gamma\left(\nu+\tfrac{1}{2}\right)\*z\left(\mathscr{C}_{\nu}\left(z\right)% \mathbf{H}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu-1}\left(z\right)\mathbf{H}_{% \nu}\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$.

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

 10.22.3 $\displaystyle\int e^{iz}z^{\nu}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=\frac{e^{iz}z^{\nu+1}}{2\nu+1}(\mathscr{C}_{\nu}\left(z\right)-i% \mathscr{C}_{\nu+1}\left(z\right)),$ $\nu\neq-\tfrac{1}{2}$, $\displaystyle\int e^{iz}z^{-\nu}\mathscr{C}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=\frac{e^{iz}z^{-\nu+1}}{1-2\nu}(\mathscr{C}_{\nu}\left(z\right)+% i\mathscr{C}_{\nu-1}\left(z\right)),$ $\nu\neq\tfrac{1}{2}$. ⓘ Symbols: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$: cylinder function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of 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 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.22(i), 10.22 and 10

### Products

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

## §10.22(ii) Integrals over Finite Intervals

Throughout this subsection $x>0$.

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

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

### Trigonometric Arguments

 10.22.13 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{2\nu}\left(2z\cos\theta\right)\cos% \left(2\mu\theta\right)\mathrm{d}\theta$ $\displaystyle=\tfrac{1}{2}\pi J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z% \right),$ $\Re\nu>-\tfrac{1}{2}$, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\Re$: real part, $n$: integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 11.4.7 (Case $\nu=n$, $\mu=0$.) Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E13 Encodings: TeX, pMML, png See also: Annotations for 10.22(ii), 10.22(ii), 10.22 and 10 10.22.14 $\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\cos\left(2\mu% \theta\right)\mathrm{d}\theta$ $\displaystyle=\pi\cos\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}% \left(z\right),$ $\Re\nu>-\tfrac{1}{2}$, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\Re$: real part, $\sin\NVar{z}$: sine function, $n$: integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 11.4.8 (Case $\nu=0$, $\mu=n$.) Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E14 Encodings: TeX, pMML, png See also: Annotations for 10.22(ii), 10.22(ii), 10.22 and 10 10.22.15 $\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\sin(2\mu\theta)% \mathrm{d}\theta$ $\displaystyle=\pi\sin(\mu\pi)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}\left(z\right),$ $\Re\nu>-1$. 10.22.16 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{0}\left(2z\sin\theta\right)\cos(2n% \theta)\mathrm{d}\theta$ $\displaystyle=\tfrac{1}{2}\pi{J_{n}^{2}}\left(z\right),$ $n=0,1,2,\ldots$.
 10.22.17 $\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos\theta\right)\cos(2\mu\theta)% \mathrm{d}\theta=\tfrac{1}{2}\pi\cot(2\nu\pi)J_{\nu+\mu}\left(z\right)J_{\nu-% \mu}\left(z\right)-\tfrac{1}{2}\pi\csc(2\nu\pi)J_{\mu-\nu}\left(z\right)J_{-% \mu-\nu}\left(z\right),$ $-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}$,
 10.22.18 $\int_{0}^{\frac{1}{2}\pi}Y_{0}\left(2z\sin\theta\right)\cos\left(2n\theta% \right)\mathrm{d}\theta=\tfrac{1}{2}\pi J_{n}\left(z\right)Y_{n}\left(z\right),$ $n=0,1,2,\dots$.
 10.22.19 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin\theta)^{\mu+1}(% \cos\theta)^{2\nu+1}\mathrm{d}\theta=2^{\nu}\Gamma\left(\nu+1\right)z^{-\nu-1}% J_{\mu+\nu+1}\left(z\right),$ $\Re\mu>-1$, $\Re\nu>-1$,
 10.22.20 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin% \theta)^{\mu}(\cos\theta)^{2\mu}\mathrm{d}\theta$ $\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\Gamma\left(\mu+\tfrac{1}{2}% \right){J_{\mu}^{2}}\left(\tfrac{1}{2}z\right),$ $\Re\mu>-\tfrac{1}{2}$, 10.22.21 $\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{\mu}\left(z\sin\theta\right)(\sin% \theta)^{\mu}(\cos\theta)^{2\mu}\mathrm{d}\theta$ $\displaystyle=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*\Gamma\left(\mu+\tfrac{1}{2}% \right)J_{\mu}\left(\tfrac{1}{2}z\right)Y_{\mu}\left(\tfrac{1}{2}z\right),$ $\Re\mu>-\tfrac{1}{2}$.
 10.22.22 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}\theta\right)J_{\nu}\left(z{% \cos^{2}}\theta\right)(\sin\theta)^{2\mu+1}(\cos\theta)^{2\nu+1}\mathrm{d}% \theta=\frac{\Gamma\left(\mu+\tfrac{1}{2}\right)\Gamma\left(\nu+\tfrac{1}{2}% \right)J_{\mu+\nu+\frac{1}{2}}\left(z\right)}{(8\pi z)^{\frac{1}{2}}\Gamma% \left(\mu+\nu+1\right)},$ $\Re\mu>-\tfrac{1}{2},\Re\nu>-\tfrac{1}{2}$.
 10.22.23 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}\theta\right)J_{% \nu}\left(z{\cos^{2}}\theta\right)(\sin\theta)^{2\alpha-1}\sec\theta\mathrm{d}\theta$ $\displaystyle=\frac{(\mu+\nu+\alpha)\Gamma\left(\mu+\alpha\right)2^{\alpha-1}}% {\nu\Gamma\left(\mu+1\right)z^{\alpha}}J_{\mu+\nu+\alpha}\left(z\right),$ $\Re(\mu+\alpha)>0$, $\Re\nu>0$. 10.22.24 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin^{2}}\theta\right)J_{% \nu}\left(z{\cos^{2}}\theta\right)\cot\theta\mathrm{d}\theta$ $\displaystyle=\tfrac{1}{2}\mu^{-1}J_{\mu+\nu}\left(z\right),$ $\Re\mu>0,\Re\nu>-1$. 10.22.25 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)I_{\nu}% \left(z\cos\theta\right)(\tan\theta)^{\mu+1}\mathrm{d}\theta$ $\displaystyle=\frac{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{% 1}{2}z)^{\mu}}{2\!\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}J_{\nu}% \left(z\right),$ $\Re\nu>\Re\mu>-1$.

For $I_{\nu}$ see §10.25(ii).

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

### Products

 10.22.27 $\displaystyle\int_{0}^{x}t{J_{\nu-1}^{2}}\left(t\right)\mathrm{d}t$ $\displaystyle=2\sum_{k=0}^{\infty}(\nu+2k){J_{\nu+2k}^{2}}\left(x\right),$ $\Re\nu>0$, 10.22.28 $\displaystyle\int_{0}^{x}t\left({J_{\nu-1}^{2}}\left(t\right)-{J_{\nu+1}^{2}}% \left(t\right)\right)\mathrm{d}t$ $\displaystyle=2\nu{J_{\nu}^{2}}\left(x\right),$ $\Re\nu>0$, 10.22.29 $\displaystyle\int_{0}^{x}t{J_{0}^{2}}\left(t\right)\mathrm{d}t$ $\displaystyle=\tfrac{1}{2}x^{2}\left({J_{0}^{2}}\left(x\right)+{J_{1}^{2}}% \left(x\right)\right).$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $x$: real variable A&S Ref: 11.3.34 Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E29 Encodings: TeX, pMML, png See also: Annotations for 10.22(ii), 10.22(ii), 10.22 and 10
 10.22.30 $\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t\right)\mathrm{d}t=\tfrac{1}{2}% \left(1-{J_{0}^{2}}\left(x\right)\right)-\sum_{k=1}^{n}{J_{k}^{2}}\left(x% \right)=\sum_{k=n+1}^{\infty}{J_{k}^{2}}\left(x\right),$ $n=0,1,2,\ldots$.

### Convolutions

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

### Fractional Integral

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

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

### Orthogonality

If $\nu>-1$, then

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

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

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

 10.22.38 $\int_{0}^{1}tJ_{\nu}\left(\alpha_{\ell}t\right)J_{\nu}\left(\alpha_{m}t\right)% \mathrm{d}t=\delta_{\ell,m}\left(\frac{a^{2}}{b^{2}}+\alpha_{\ell}^{2}-\nu^{2}% \right)\frac{(J_{\nu}\left(\alpha_{\ell}\right))^{2}}{2\alpha_{\ell}^{2}},$

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

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

When $x>0$

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

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

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

 10.22.41 $\displaystyle\int_{0}^{\infty}J_{\nu}\left(t\right)\mathrm{d}t$ $\displaystyle=1,$ $\Re\nu>-1$, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\Re$: real part and $\nu$: complex parameter A&S Ref: 11.4.17 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E41 Encodings: TeX, pMML, png See also: Annotations for 10.22(iv), 10.22 and 10 10.22.42 $\displaystyle\int_{0}^{\infty}Y_{\nu}\left(t\right)\mathrm{d}t$ $\displaystyle=-\tan\left(\tfrac{1}{2}\nu\pi\right),$ $|\Re\nu|<1$.
 10.22.43 $\displaystyle\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)\mathrm{d}t$ $\displaystyle=2^{\mu}\frac{\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{% 1}{2}\right)}{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},$ $\Re(\mu+\nu)>-1$, $\Re\mu<\tfrac{1}{2}$, 10.22.44 $\displaystyle\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)\mathrm{d}t$ $\displaystyle=\frac{2^{\mu}}{\pi}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+% \tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}% \right)\sin\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi,$ $\Re(\mu\pm\nu)>-1$, $\Re\mu<\tfrac{1}{2}$.
 10.22.45 $\int_{0}^{\infty}\frac{1-J_{0}\left(t\right)}{t^{\mu}}\mathrm{d}t=-\frac{\pi% \sec\left(\frac{1}{2}\mu\pi\right)}{2^{\mu}{\Gamma^{2}}\left(\frac{1}{2}\mu+% \frac{1}{2}\right)},$ $1<\Re\mu<3$.
 10.22.46 $\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at\right)}{(t^{2}+b^{2})^{\mu+1}}% \mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\Gamma\left(\mu+1\right)}K_{\nu-% \mu}\left(ab\right),$ $a>0$, $\Re b>0$, $-1<\Re\nu<2\Re\mu+\tfrac{3}{2}$.
 10.22.47 $\int_{0}^{\infty}\frac{t^{\nu}Y_{\nu}\left(at\right)}{t^{2}+b^{2}}\mathrm{d}t=% -b^{\nu-1}K_{\nu}\left(ab\right),$ $a>0,\Re b>0,-\tfrac{1}{2}<\Re\nu<\tfrac{5}{2}$. ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the second kind, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\Re$: real part and $\nu$: complex parameter A&S Ref: 11.4.46 (is the special case $\nu=0$.) Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E47 Encodings: TeX, pMML, png See also: Annotations for 10.22(iv), 10.22 and 10

For $K_{\nu}$ see §10.25(ii).

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

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

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

For $I$ and $K$ see §10.25(ii).

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

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

### Orthogonality

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

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

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

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

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

When $\Re\mu>-1$

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

When $\Re\mu>0$,

 10.22.61 $\int_{0}^{\infty}t^{-1}e^{ibt}J_{\mu}\left(at\right)\mathrm{d}t=\begin{cases}(% 1/\mu)\exp\left(i\mu\operatorname{arcsin}\left(b/a\right)\right),&0\leq b\leq a% ,\\ \dfrac{a^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{\mu\left(b+(b^{2}-a^{2})^{% \frac{1}{2}}\right)^{\mu}},&0

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

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

When $\Re\mu>0$,

 10.22.63 $\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\mu-1}\left(bt\right)\mathrm{d}t=% \begin{cases}b^{\mu-1}a^{-\mu},&00),\\ 0,&0 ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral A&S Ref: 11.4.42 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E63 Encodings: TeX, pMML, png See also: Annotations for 10.22(iv), 10.22(iv), 10.22 and 10

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

 10.22.64 $\int_{0}^{\infty}J_{\mu+2n+1}\left(at\right)J_{\mu}\left(bt\right)\mathrm{d}t=% \begin{cases}\dfrac{b^{\mu}\Gamma\left(\mu+n+1\right)}{a^{\mu+1}n!}\mathbf{F}% \left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}\right),&00),\\ 0,&0
 10.22.65 $\int_{0}^{\infty}J_{0}\left(at\right)\left(J_{0}\left(bt\right)-J_{0}\left(ct% \right)\right)\frac{\mathrm{d}t}{t}=\begin{cases}0,&0\leq b ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\ln\NVar{z}$: principal branch of logarithm function A&S Ref: 11.4.43 (Case $b=0$.) Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E65 Encodings: TeX, pMML, png See also: Annotations for 10.22(iv), 10.22(iv), 10.22 and 10

### Other Double Products

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

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

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

 10.22.69 $\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}\left(bt\right)% \frac{t\mathrm{d}t}{t^{2}-z^{2}}$ $\displaystyle=\left\{\begin{array}[]{ll}\frac{1}{2}\pi iJ_{\nu}\left(bz\right)% {H^{(1)}_{\nu}}\left(az\right),&a>b\\ \frac{1}{2}\pi iJ_{\nu}\left(az\right){H^{(1)}_{\nu}}\left(bz\right),&b>a\end{% array}\right\},$ $\Re\nu>-1,\Im z>0$. 10.22.70 $\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu+1}\left(bt\right)% \frac{t\mathrm{d}t}{t^{2}-z^{2}}$ $\displaystyle=\frac{1}{2}\pi J_{\nu+1}\left(bz\right){H^{(1)}_{\nu}}\left(az% \right),$ $a\geq b>0$, $\Re\nu>-\tfrac{3}{2},\Im z>0$.

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

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

### Triple Products

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

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

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

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

 10.22.73 $\displaystyle A$ $\displaystyle=s(s-a)(s-b)(s-c),$ $\displaystyle s$ $\displaystyle=\tfrac{1}{2}(a+b+c).$ ⓘ 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), 10.22(iv), 10.22 and 10

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

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

 10.22.74 $\displaystyle\int_{0}^{\infty}J_{\nu}\left(at\right)J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)t^{1-\nu}\mathrm{d}t$ $\displaystyle=\begin{cases}\dfrac{2^{\nu-1}A^{\nu-\frac{1}{2}}}{\pi^{\frac{1}{% 2}}(abc)^{\nu}\Gamma\left(\nu+\frac{1}{2}\right)},&A>0,\\ 0,&A\leq 0.\end{cases}$ If $|\nu|<\tfrac{1}{2}$, then 10.22.75 $\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)t^{1+\nu}\mathrm{d}t$ $\displaystyle=\begin{cases}-\dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{% \frac{1}{2}}2^{\nu+1}\Gamma\left(\frac{1}{2}-\nu\right)},&0b+c.\end{cases}$

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

Hankel’s inversion theorem is given by

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

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 $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, and ${H^{(2)}_{\nu}}\left(z\right)$, including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (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).