# §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 Ch.10
 10.22.2 $\int z^{\nu}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z=\pi^{\frac{1}{2}}2^{% \nu-1}\Gamma\left(\nu+\tfrac{1}{2}\right)\*z\left(\mathscr{C}_{\nu}\left(z% \right)\mathbf{H}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu-1}\left(z\right)% \mathbf{H}_{\nu}\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$.

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

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

### 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 $\int_{0}^{x}t^{\mu}J_{\nu}\left(t\right)\,\mathrm{d}t=x^{\mu}\frac{\Gamma\left% (\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-% \frac{1}{2}\mu+\frac{1}{2}\right)}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\Gamma% \left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k\right)}{\Gamma\left(\frac{1}% {2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}J_{\nu+2k+1}\left(x\right),$ $\Re\left(\mu+\nu+1\right)>0$.
 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 $\displaystyle x\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\,\mathrm{d}t$ $\displaystyle=2\sum_{k=0}^{\infty}(2k+3)(\psi\left(k+2\right)-\psi\left(1% \right))J_{2k+3}\left(x\right)=x-2J_{1}\left(x\right)+2\sum_{k=0}^{\infty}(2k+% 5)\*(\psi\left(k+3\right)-\psi\left(1\right)-1)J_{2k+5}\left(x\right),$

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 Ch.10 10.22.14 $\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\cos\left(2\mu% \theta\right)\,\mathrm{d}\theta$ $\displaystyle=\pi\cos\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}% \left(z\right),$ $\Re\nu>-\tfrac{1}{2}$, ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\Re$: real part, $\sin\NVar{z}$: sine function, $n$: integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 11.4.8 (Case $\nu=0$, $\mu=n$.) Referenced by: §10.22(ii) Permalink: http://dlmf.nist.gov/10.22.E14 Encodings: TeX, pMML, png See also: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10 10.22.15 $\displaystyle\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\sin\left(2\mu% \theta\right)\,\mathrm{d}\theta$ $\displaystyle=\pi\sin\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu}% \left(z\right),$ $\Re\nu>-1$. 10.22.16 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{0}\left(2z\sin\theta\right)\cos\left(% 2n\theta\right)\,\mathrm{d}\theta$ $\displaystyle=\tfrac{1}{2}\pi{J_{n}}^{2}\left(z\right),$ $n=0,1,2,\dotsc$.
 10.22.17 $\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos\theta\right)\cos\left(2\mu\theta% \right)\,\mathrm{d}\theta=\tfrac{1}{2}\pi\cot\left(2\nu\pi\right)J_{\nu+\mu}% \left(z\right)J_{\nu-\mu}\left(z\right)-\tfrac{1}{2}\pi\csc\left(2\nu\pi\right% )J_{\mu-\nu}\left(z\right)J_{-\mu-\nu}\left(z\right),$ $-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}$,
 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 $\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin}^{2}\theta\right)J_{% \nu}\left(z{\cos}^{2}\theta\right)(\sin\theta)^{2\alpha-1}\sec\theta\,\mathrm{% d}\theta$ $\displaystyle=\frac{(\mu+\nu+\alpha)\Gamma\left(\mu+\alpha\right)2^{\alpha-1}}% {\nu\Gamma\left(\mu+1\right)z^{\alpha}}J_{\mu+\nu+\alpha}\left(z\right),$ $\Re\left(\mu+\alpha\right)>0$, $\Re\nu>0$. 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 Ch.10
 10.22.30 $\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}% \left(1-{J_{0}}^{2}\left(x\right)\right)-\sum_{k=1}^{n}{J_{k}}^{2}\left(x% \right)=\sum_{k=n+1}^{\infty}{J_{k}}^{2}\left(x\right),$ $n=0,1,2,\dotsc$.

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

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

### Orthogonality

If $\nu>-1$, then

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

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

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

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

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

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

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)). Compare (10.22.11) and (10.22.12).

## §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 Ch.10 10.22.42 $\displaystyle\int_{0}^{\infty}Y_{\nu}\left(t\right)\,\mathrm{d}t$ $\displaystyle=-\tan\left(\tfrac{1}{2}\nu\pi\right),$ $|\Re\nu|<1$.
 10.22.43 $\displaystyle\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)\,\mathrm{d}t$ $\displaystyle=2^{\mu}\frac{\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{% 1}{2}\right)}{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},$ $\Re\left(\mu+\nu\right)>-1$, $\Re\mu<\tfrac{1}{2}$, ⓘ 10.22.44 $\displaystyle\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)\,\mathrm{d}t$ $\displaystyle=\frac{2^{\mu}}{\pi}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+% \tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}% \right)\sin\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi,$ $\Re\left(\mu\pm\nu\right)>-1$, $\Re\mu<\tfrac{1}{2}$.
 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 Keywords: Mellin transform A&S Ref: 11.4.46 (is the special case $\nu=0$.) Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E47 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22 and Ch.10

For $K_{\nu}$ see §10.25(ii).

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

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

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

For $I$ and $K$ see §10.25(ii).

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

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, $\Re\left(\mu+\nu+1\right)>\Re\lambda>-1$.

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

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

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 Keywords: Mellin transform A&S Ref: 11.4.41 Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.22.E62 Encodings: TeX, pMML, png See also: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\mu>0$,

 10.22.63 $\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\mu-1}\left(bt\right)\,\mathrm{d}t=% \begin{cases}b^{\mu-1}a^{-\mu},&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 Ch.10

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

 10.22.64 $\int_{0}^{\infty}J_{\mu+2n+1}\left(at\right)J_{\mu}\left(bt\right)\,\mathrm{d}% t=\begin{cases}\dfrac{b^{\mu}\Gamma\left(\mu+n+1\right)}{a^{\mu+1}n!}\mathbf{F% }\left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}\right),&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 Ch.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 $\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{\nu}\left(at% \right)J_{\nu}\left(bt\right)\,\mathrm{d}t$ $\displaystyle=\frac{1}{2p^{2}}\exp\left(-\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{% \nu}\left(\frac{ab}{2p^{2}}\right),$ $\Re\nu>-1,\Re\left(p^{2}\right)>0$. 10.22.68 $\displaystyle\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{0}\left(at\right% )Y_{0}\left(at\right)\,\mathrm{d}t$ $\displaystyle=-\frac{1}{2\pi p^{2}}\exp\left(-\frac{a^{2}}{2p^{2}}\right)K_{0}% \left(\frac{a^{2}}{2p^{2}}\right),$ $\Re\left(p^{2}\right)>0$.

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

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

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

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

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

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

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

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

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

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

 10.22.79 $f(x)=\int_{0}^{\infty}(xt)^{\frac{1}{2}}\frac{cJ_{\nu}\left(xt\right)+t^{2\nu}% J_{-\nu}\left(xt\right)}{c^{2}+2c\cos\left(\nu\pi\right)t^{2\nu}+t^{4\nu}}\*% \int_{0}^{\infty}(yt)^{\frac{1}{2}}\left(cJ_{\nu}\left(yt\right)+t^{2\nu}J_{-% \nu}\left(yt\right)\right)f(y)\,\mathrm{d}y\,\mathrm{d}t,$ $0<\nu<1,c>0$.

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

## §10.22(vi) Compendia

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