10 Bessel Functions Bessel and Hankel Functions 10.21 Zeros 10.23 Sums

§10.22 Integrals

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

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

§10.22(i) Indefinite Integrals

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

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

10.22.1

$\int z^{\nu+1}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z=z^{\nu+1}\mathscr{C% }_{\nu+1}\left(z\right),$

$\int z^{-\nu+1}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z=-z^{-\nu+1}% \mathscr{C}_{\nu-1}\left(z\right).$

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Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : 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 $\int z^{\nu}\mathscr{C}_{\nu}\left(z\right)\,\mathrm{d}z=\pi^{\frac{1}{2}}2^{% \nu-1}\Gamma\left(\nu+\tfrac{1}{2}\right)\*z\left(\mathscr{C}_{\nu}\left(z% \right)\mathbf{H}_{\nu-1}\left(z\right)-\mathscr{C}_{\nu-1}\left(z\right)% \mathbf{H}_{\nu}\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$ .

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

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

10.22.3

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

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

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

Products

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

10.22.4 $\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(bz)\,\mathrm{d}z=\frac% {z\left(a\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\mu}(bz)-b\mathscr{C}_% {\mu}\left(az\right)\mathscr{D}_{\mu+1}(bz)\right)}{a^{2}-b^{2}},$ $a^{2}\neq b^{2}$ ,

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Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : complex variable and $\mathscr{D}_{\nu}(z)$ : cylinder function
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

10.22.5 $\int z\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)\,\mathrm{d}z=% \tfrac{1}{4}z^{2}\left(2\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\mu}(az)-% \mathscr{C}_{\mu-1}\left(az\right)\mathscr{D}_{\mu+1}(az)-\mathscr{C}_{\mu+1}% \left(az\right)\mathscr{D}_{\mu-1}(az)\right),$

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Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : complex variable and $\mathscr{D}_{\nu}(z)$ : cylinder function
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

10.22.6 $\int\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)\frac{\,\mathrm{d}z}{% z}=-\frac{az(\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{\nu}(az)-\mathscr{% C}_{\mu}\left(az\right)\mathscr{D}_{\nu+1}(az))}{\mu^{2}-\nu^{2}}+\frac{% \mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)}{\mu+\nu},$ $\mu^{2}\neq\nu^{2}$ ,

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Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function
Permalink:: http://dlmf.nist.gov/10.22.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

10.22.7

$\int z^{\mu+\nu+1}\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)\,% \mathrm{d}z=\frac{z^{\mu+\nu+2}}{2(\mu+\nu+1)}\*\left(\mathscr{C}_{\mu}\left(% az\right)\mathscr{D}_{\nu}(az)+\mathscr{C}_{\mu+1}\left(az\right)\mathscr{D}_{% \nu+1}(az)\right),$ $\mu+\nu\neq-1$ ,

$\int z^{-\mu-\nu+1}\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)\,% \mathrm{d}z=\frac{z^{-\mu-\nu+2}}{2(1-\mu-\nu)}\*\left(\mathscr{C}_{\mu}\left(% az\right)\mathscr{D}_{\nu}(az)+\mathscr{C}_{\mu-1}\left(az\right)\mathscr{D}_{% \nu-1}(az)\right),$ $\mu+\nu\neq 1$ .

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

§10.22(ii) Integrals over Finite Intervals

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

Throughout this subsection .

10.22.8 $\int_{0}^{x}J_{\nu}\left(t\right)\,\mathrm{d}t=2\sum_{k=0}^{\infty}J_{\nu+2k+1% }\left(x\right),$ $\Re\nu>-1$ .

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

10.22.9 $\int_{0}^{x}J_{2n}\left(t\right)\,\mathrm{d}t=\int_{0}^{x}J_{0}\left(t\right)% \,\mathrm{d}t-2\sum_{k=0}^{n-1}J_{2k+1}\left(x\right),\quad\int_{0}^{x}J_{2n+1% }\left(t\right)\,\mathrm{d}t=1-J_{0}\left(x\right)-2\sum_{k=1}^{n}J_{2k}\left(% x\right),$ $n=0,1,\dots$ .

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, : nonnegative integer and : real variable
A&S Ref:: 11.1.3, 11.1.4
Permalink:: http://dlmf.nist.gov/10.22.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.10 $\int_{0}^{x}t^{\mu}J_{\nu}\left(t\right)\,\mathrm{d}t=x^{\mu}\frac{\Gamma\left% (\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-% \frac{1}{2}\mu+\frac{1}{2}\right)}\*\sum_{k=0}^{\infty}\frac{(\nu+2k+1)\Gamma% \left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}+k\right)}{\Gamma\left(\frac{1}% {2}\nu+\frac{1}{2}\mu+\frac{3}{2}+k\right)}J_{\nu+2k+1}\left(x\right),$ $\Re\left(\mu+\nu+1\right)>0$ .

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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 , $\int$ : integral, $\Re$ : real part, : nonnegative integer, : real variable and $\nu$ : complex parameter
A&S Ref:: 11.1.1
Permalink:: http://dlmf.nist.gov/10.22.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.11 $\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\,\mathrm{d}t=\frac{1}{2}\sum_{k=1}% ^{\infty}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{2}x)^{k}% J_{k}\left(x\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\int$ : integral, : nonnegative integer and : real variable
Referenced by:: §10.22(iii), §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.22.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

10.22.12 $x\int_{0}^{x}\frac{1-J_{0}\left(t\right)}{t}\,\mathrm{d}t=2\sum_{k=0}^{\infty}% (2k+3)(\psi\left(k+2\right)-\psi\left(1\right))J_{2k+3}\left(x\right)=x-2J_{1}% \left(x\right)+2\sum_{k=0}^{\infty}(2k+5)\*(\psi\left(k+3\right)-\psi\left(1% \right)-1)J_{2k+5}\left(x\right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\int$ : integral, : nonnegative integer and : real variable
A&S Ref:: 11.1.19
Referenced by:: §10.22(ii), §10.22(iii), §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.22.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22 and Ch.10

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

Trigonometric Arguments

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

10.22.13 $\int_{0}^{\frac{1}{2}\pi}J_{2\nu}\left(2z\cos\theta\right)\cos\left(2\mu\theta% \right)\,\mathrm{d}\theta=\tfrac{1}{2}\pi J_{\nu+\mu}\left(z\right)J_{\nu-\mu}% \left(z\right),$ $\Re\nu>-\tfrac{1}{2}$ ,

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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 , $\int$ : integral, $\Re$ : real part, : integer, : 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 $\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\cos\left(2\mu\theta\right)\,% \mathrm{d}\theta=\pi\cos\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu% }\left(z\right),$ $\Re\nu>-\tfrac{1}{2}$ ,

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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 , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : integer, : 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 $\int_{0}^{\pi}J_{2\nu}\left(2z\sin\theta\right)\sin\left(2\mu\theta\right)\,% \mathrm{d}\theta=\pi\sin\left(\mu\pi\right)J_{\nu+\mu}\left(z\right)J_{\nu-\mu% }\left(z\right),$ $\Re\nu>-1$ .

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.16 $\int_{0}^{\frac{1}{2}\pi}J_{0}\left(2z\sin\theta\right)\cos\left(2n\theta% \right)\,\mathrm{d}\theta=\tfrac{1}{2}\pi{J_{n}}^{2}\left(z\right),$ $n=0,1,2,\dotsc$ .

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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 , $\int$ : integral, $\sin\NVar{z}$ : sine function, : integer and : complex variable
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.17 $\int_{0}^{\frac{1}{2}\pi}Y_{2\nu}\left(2z\cos\theta\right)\cos\left(2\mu\theta% \right)\,\mathrm{d}\theta=\tfrac{1}{2}\pi\cot\left(2\nu\pi\right)J_{\nu+\mu}% \left(z\right)J_{\nu-\mu}\left(z\right)-\tfrac{1}{2}\pi\csc\left(2\nu\pi\right% )J_{\mu-\nu}\left(z\right)J_{-\mu-\nu}\left(z\right),$ $-\tfrac{1}{2}<\Re\nu<\tfrac{1}{2}$ ,

10.22.18 $\int_{0}^{\frac{1}{2}\pi}Y_{0}\left(2z\sin\theta\right)\cos\left(2n\theta% \right)\,\mathrm{d}\theta=\tfrac{1}{2}\pi J_{n}\left(z\right)Y_{n}\left(z% \right),$ $n=0,1,2,\dots$ .

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function, : integer and : complex variable
A&S Ref:: 11.4.9
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.19 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin\theta)^{\mu+1}(% \cos\theta)^{2\nu+1}\,\mathrm{d}\theta=2^{\nu}\Gamma\left(\nu+1\right)z^{-\nu-% 1}J_{\mu+\nu+1}\left(z\right),$ $\Re\mu>-1$ , $\Re\nu>-1$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 11.4.10
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.20 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)(\sin\theta)^{\mu}(% \cos\theta)^{2\mu}\,\mathrm{d}\theta=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*% \Gamma\left(\mu+\tfrac{1}{2}\right){J_{\mu}}^{2}\left(\tfrac{1}{2}z\right),$ $\Re\mu>-\tfrac{1}{2}$ ,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and : complex variable
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.21 $\int_{0}^{\frac{1}{2}\pi}Y_{\mu}\left(z\sin\theta\right)(\sin\theta)^{\mu}(% \cos\theta)^{2\mu}\,\mathrm{d}\theta=\pi^{\frac{1}{2}}2^{\mu-1}z^{-\mu}\*% \Gamma\left(\mu+\tfrac{1}{2}\right)J_{\mu}\left(\tfrac{1}{2}z\right)Y_{\mu}% \left(\tfrac{1}{2}z\right),$ $\Re\mu>-\tfrac{1}{2}$ .

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and : complex variable
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.22 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin}^{2}\theta\right)J_{\nu}\left(z{% \cos}^{2}\theta\right)(\sin\theta)^{2\mu+1}(\cos\theta)^{2\nu+1}\,\mathrm{d}% \theta=\frac{\Gamma\left(\mu+\tfrac{1}{2}\right)\Gamma\left(\nu+\tfrac{1}{2}% \right)J_{\mu+\nu+\frac{1}{2}}\left(z\right)}{(8\pi z)^{\frac{1}{2}}\Gamma% \left(\mu+\nu+1\right)},$ $\Re\mu>-\tfrac{1}{2},\Re\nu>-\tfrac{1}{2}$ .

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.23 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin}^{2}\theta\right)J_{\nu}\left(z{% \cos}^{2}\theta\right)(\sin\theta)^{2\alpha-1}\sec\theta\,\mathrm{d}\theta=% \frac{(\mu+\nu+\alpha)\Gamma\left(\mu+\alpha\right)2^{\alpha-1}}{\nu\Gamma% \left(\mu+1\right)z^{\alpha}}J_{\mu+\nu+\alpha}\left(z\right),$ $\Re\left(\mu+\alpha\right)>0$ , $\Re\nu>0$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function, : complex variable, $\nu$ : complex parameter and $\alpha$ : positive integer
A&S Ref:: 11.4.11
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.24 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z{\sin}^{2}\theta\right)J_{\nu}\left(z{% \cos}^{2}\theta\right)\cot\theta\,\mathrm{d}\theta=\tfrac{1}{2}\mu^{-1}J_{\mu+% \nu}\left(z\right),$ $\Re\mu>0,\Re\nu>-1$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.25 $\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta\right)I_{\nu}\left(z\cos% \theta\right)(\tan\theta)^{\mu+1}\,\mathrm{d}\theta=\frac{\Gamma\left(\tfrac{1% }{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{1}{2}z)^{\mu}}{2\Gamma\left(\tfrac{1}{2}% \nu+\tfrac{1}{2}\mu+1\right)}J_{\nu}\left(z\right),$ $\Re\nu>\Re\mu>-1$ .

For $I_{\nu}$ see §10.25(ii).

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

ⓘ

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 , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

Products

ⓘ

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

10.22.27 $\int_{0}^{x}t{J_{\nu-1}}^{2}\left(t\right)\,\mathrm{d}t=2\sum_{k=0}^{\infty}(% \nu+2k){J_{\nu+2k}}^{2}\left(x\right),$ $\Re\nu>0$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, : nonnegative integer, : real variable and $\nu$ : complex parameter
A&S Ref:: 11.3.32
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.28 $\int_{0}^{x}t\left({J_{\nu-1}}^{2}\left(t\right)-{J_{\nu+1}}^{2}\left(t\right)% \right)\,\mathrm{d}t=2\nu{J_{\nu}}^{2}\left(x\right),$ $\Re\nu>0$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, : real variable and $\nu$ : complex parameter
A&S Ref:: 11.3.33
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.29 $\int_{0}^{x}t{J_{0}}^{2}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}x^{2}\left({J_% {0}}^{2}\left(x\right)+{J_{1}}^{2}\left(x\right)\right).$

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : real variable
A&S Ref:: 11.3.34
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.30 $\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}% \left(1-{J_{0}}^{2}\left(x\right)\right)-\sum_{k=1}^{n}{J_{k}}^{2}\left(x% \right)=\sum_{k=n+1}^{\infty}{J_{k}}^{2}\left(x\right),$ $n=0,1,2,\dotsc$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer, : nonnegative integer and : real variable
A&S Ref:: 11.3.35
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E30
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

Convolutions

ⓘ

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

10.22.31 $\int_{0}^{x}J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\,\mathrm{d}t=2\sum_{k% =0}^{\infty}(-1)^{k}J_{\mu+\nu+2k+1}\left(x\right),$ $\Re\mu>-1,\Re\nu>-1$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, : nonnegative integer, : real variable and $\nu$ : complex parameter
A&S Ref:: 11.3.37
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.32 $\int_{0}^{x}J_{\nu}\left(t\right)J_{1-\nu}\left(x-t\right)\,\mathrm{d}t=J_{0}% \left(x\right)-\cos x,$ $-1<\Re\nu<2$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, : real variable and $\nu$ : complex parameter
A&S Ref:: 11.3.38
Permalink:: http://dlmf.nist.gov/10.22.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.33 $\int_{0}^{x}J_{\nu}\left(t\right)J_{-\nu}\left(x-t\right)\,\mathrm{d}t=\sin x,$ $|\Re\nu|<1$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : real variable and $\nu$ : complex parameter
A&S Ref:: 11.3.39
Permalink:: http://dlmf.nist.gov/10.22.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

10.22.34 $\int_{0}^{x}t^{-1}J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\,\mathrm{d}t=% \frac{J_{\mu+\nu}\left(x\right)}{\mu},$ $\Re\mu>0,\Re\nu>-1$ .

ⓘ

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

10.22.35 $\int_{0}^{x}\frac{J_{\mu}\left(t\right)J_{\nu}\left(x-t\right)\,\mathrm{d}t}{t% (x-t)}=\frac{(\mu+\nu)J_{\mu+\nu}\left(x\right)}{\mu\nu x},$ $\Re\mu>0,\Re\nu>0$ .

ⓘ

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

Fractional Integral

ⓘ

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

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\Re$ : real part, : nonnegative integer, : 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 th repeated integral of $J_{\nu}\left(x\right)$ (§§1.4(v) and 1.15(vi)).

Orthogonality

ⓘ

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

If $\nu>-1$ , then

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

ⓘ

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

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

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

ⓘ

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

When

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\gamma$ : Euler’s constant, $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : real variable
A&S Ref:: 11.1.20
Referenced by:: §10.22(ii), §10.22(iii), §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.22.E39
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iii), §10.22 and Ch.10

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

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : real variable
A&S Ref:: 11.1.21
Referenced by:: §10.22(iii), §10.43(ii)
Permalink:: http://dlmf.nist.gov/10.22.E40
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iii), §10.22 and Ch.10

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

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

ⓘ

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

10.22.41 $\int_{0}^{\infty}J_{\nu}\left(t\right)\,\mathrm{d}t=1,$ $\Re\nu>-1$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : 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 $\int_{0}^{\infty}Y_{\nu}\left(t\right)\,\mathrm{d}t=-\tan\left(\tfrac{1}{2}\nu% \pi\right),$ $|\Re\nu|<1$ .

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\tan\NVar{z}$ : tangent function and $\nu$ : complex parameter
A&S Ref:: 11.4.20, 11.4.21
Permalink:: http://dlmf.nist.gov/10.22.E42
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.43 $\int_{0}^{\infty}t^{\mu}J_{\nu}\left(t\right)\,\mathrm{d}t=2^{\mu}\frac{\Gamma% \left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{% 1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}\right)},$ $\Re\left(\mu+\nu\right)>-1$ , $\Re\mu<\tfrac{1}{2}$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter
Keywords:: Mellin transform
A&S Ref:: 11.4.16
Referenced by:: §10.22(iii), §10.59
Permalink:: http://dlmf.nist.gov/10.22.E43
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.44 $\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)\,\mathrm{d}t=\frac{2^{\mu}}{\pi}% \Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\Gamma\left(% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\sin\left(\tfrac{1}{2}\mu-% \tfrac{1}{2}\nu\right)\pi,$ $\Re\left(\mu\pm\nu\right)>-1$ , $\Re\mu<\tfrac{1}{2}$ .

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $\nu$ : complex parameter
Keywords:: Mellin transform
A&S Ref:: 11.4.19 (modified)
Referenced by:: §10.22(iii)
Permalink:: http://dlmf.nist.gov/10.22.E44
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.45 $\int_{0}^{\infty}\frac{1-J_{0}\left(t\right)}{t^{\mu}}\,\mathrm{d}t=-\frac{\pi% \sec\left(\frac{1}{2}\mu\pi\right)}{2^{\mu}{\Gamma}^{2}\left(\frac{1}{2}\mu+% \frac{1}{2}\right)},$ $1<\Re\mu<3$ .

ⓘ

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

10.22.46 $\int_{0}^{\infty}\frac{t^{\nu+1}J_{\nu}\left(at\right)}{(t^{2}+b^{2})^{\mu+1}}% \,\mathrm{d}t=\frac{a^{\mu}b^{\nu-\mu}}{2^{\mu}\Gamma\left(\mu+1\right)}K_{\nu% -\mu}\left(ab\right),$

, $\Re b>0$ , $-1<\Re\nu<2\Re\mu+\tfrac{3}{2}$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and $\nu$ : complex parameter
Keywords:: Mellin transform
A&S Ref:: 11.4.44
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E46
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.47 $\int_{0}^{\infty}\frac{t^{\nu}Y_{\nu}\left(at\right)}{t^{2}+b^{2}}\,\mathrm{d}% t=-b^{\nu-1}K_{\nu}\left(ab\right),$ $a>0,\Re b>0,-\tfrac{1}{2}<\Re\nu<\tfrac{5}{2}$ .

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and $\nu$ : complex parameter
Keywords:: Mellin transform
A&S Ref:: 11.4.46 (is the special case $\nu=0$ .)
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E47
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

For $K_{\nu}$ see §10.25(ii).

10.22.48 $\int_{0}^{\infty}J_{\mu}\left(x\cosh\phi\right)(\cosh\phi)^{1-\mu}(\sinh\phi)^% {2\nu+1}\,\mathrm{d}\phi=2^{\nu}\Gamma\left(\nu+1\right)x^{-\nu-1}J_{\mu-\nu-1% }\left(x\right),$ $x>0,\Re\nu>-1,\Re\mu>2\Re\nu+\tfrac{1}{2}$ .

ⓘ

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

10.22.49 $\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt\right)\,\mathrm{d}t=\frac{(% \tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+\nu\right)\*\mathbf{F}\left(% \frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac{b^{2}}{a^{2}}\right),$ $\Re\left(\mu+\nu\right)>0,\Re\left(a\pm ib\right)>0$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter
Keywords:: Laplace transform, Mellin transform
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E49
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

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

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.22.E50
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

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

10.22.51 $\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)t^{\nu+1}\,% \mathrm{d}t=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp\left(-\frac{b^{2}}{4p^{2}}% \right),$ $\Re\nu>-1$ , $\Re\left(p^{2}\right)>0$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter
A&S Ref:: 11.4.29
Permalink:: http://dlmf.nist.gov/10.22.E51
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

10.22.52 $\int_{0}^{\infty}J_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}\right)\,\mathrm{d% }t=\frac{\sqrt{\pi}}{2p}\exp\left(-\frac{b^{2}}{8p^{2}}\right)I_{\ifrac{\nu}{2% }}\left(\frac{b^{2}}{8p^{2}}\right),$ $\Re\nu>-1,\Re\left(p^{2}\right)>0$ ,

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.22.E52
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

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

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, $\sec\NVar{z}$ : secant function, $\tan\NVar{z}$ : tangent function and $\nu$ : complex parameter
A&S Ref:: 11.4.30
Permalink:: http://dlmf.nist.gov/10.22.E53
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

For and see §10.25(ii).

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter
A&S Ref:: 11.4.28 (modified)
Permalink:: http://dlmf.nist.gov/10.22.E54
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22 and Ch.10

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

Orthogonality

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

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : integer and $\nu$ : complex parameter
A&S Ref:: 11.4.6
Referenced by:: §10.22(ii)
Permalink:: http://dlmf.nist.gov/10.22.E55
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

ⓘ

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

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

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter
Keywords:: Mellin transform
A&S Ref:: 11.4.33, 11.4.34
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E56
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

If , then interchange and , and also $\mu$ and $\nu$ . If , then

10.22.57 $\int_{0}^{\infty}\frac{J_{\mu}\left(at\right)J_{\nu}\left(at\right)}{t^{% \lambda}}\,\mathrm{d}t=\frac{(\frac{1}{2}a)^{\lambda-1}\Gamma\left(\frac{1}{2}% \mu+\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac{1}{2}\right)\Gamma\left(\lambda% \right)}{2\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1% }{2}\right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{% 1}{2}\right)\Gamma\left(\frac{1}{2}\lambda+\frac{1}{2}\mu+\frac{1}{2}\nu+\frac% {1}{2}\right)},$ $\Re\left(\mu+\nu+1\right)>\Re\lambda>0$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part and $\nu$ : complex parameter
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/10.22.E57
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $\nu$ : complex parameter
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/10.22.E58
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\mu>-1$

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{arcsin}\NVar{z}$ : arcsine function
A&S Ref:: 11.4.37, 11.4.38
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E59
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $\ln\NVar{z}$ : principal branch of logarithm function
A&S Ref:: 11.4.40
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E60
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\mu>0$ ,

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{arcsin}\NVar{z}$ : arcsine function
Keywords:: Mellin transform
A&S Ref:: 11.4.35, 11.4.36
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E61
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

10.22.62 $\int_{0}^{\infty}t^{\mu-\nu+1}J_{\mu}\left(at\right)J_{\nu}\left(bt\right)\,% \mathrm{d}t=\begin{cases}0,&0<b<a,\\ \dfrac{2^{\mu-\nu+1}a^{\mu}(b^{2}-a^{2})^{\nu-\mu-1}}{b^{\nu}\Gamma\left(\nu-% \mu\right)},&0<a\leq b.\end{cases}$

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\nu$ : complex parameter
Keywords:: Mellin transform
A&S Ref:: 11.4.41
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E62
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

When $\Re\mu>0$ ,

10.22.63 $\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\mu-1}\left(bt\right)\,\mathrm{d}t=% \begin{cases}b^{\mu-1}a^{-\mu},&0<b<a,\\ (2b)^{-1},&b=a(>0),\\ 0,&0<a<b.\end{cases}$

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of and $\int$ : integral
A&S Ref:: 11.4.42
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E63
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

10.22.64 $\int_{0}^{\infty}J_{\mu+2n+1}\left(at\right)J_{\mu}\left(bt\right)\,\mathrm{d}% t=\begin{cases}\dfrac{b^{\mu}\Gamma\left(\mu+n+1\right)}{a^{\mu+1}n!}\mathbf{F% }\left(-n,\mu+n+1;\mu+1;\dfrac{b^{2}}{a^{2}}\right),&0<b<a,\\ (-1)^{n}/(2a),&b=a(>0),\\ 0,&0<a<b.\end{cases}$

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and : integer
Referenced by:: §10.22(iv), §10.59
Permalink:: http://dlmf.nist.gov/10.22.E64
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

10.22.65 $\int_{0}^{\infty}J_{0}\left(at\right)\left(J_{0}\left(bt\right)-J_{0}\left(ct% \right)\right)\frac{\,\mathrm{d}t}{t}=\begin{cases}0,&0\leq b<a,0<c\leq a,\\ \ln\left(c/a\right),&0\leq b<a\leq c.\end{cases}$

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function
A&S Ref:: 11.4.43 (Case .)
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E65
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

Other Double Products

ⓘ

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

In (10.22.66)–(10.22.70) are positive constants.

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind and $\nu$ : complex parameter
Keywords:: Laplace transform
Referenced by:: §10.22(iv), §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E66
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

10.22.67 $\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{\nu}\left(at\right)J_{\nu}% \left(bt\right)\,\mathrm{d}t=\frac{1}{2p^{2}}\exp\left(-\frac{a^{2}+b^{2}}{4p^% {2}}\right)I_{\nu}\left(\frac{ab}{2p^{2}}\right),$ $\Re\nu>-1,\Re\left(p^{2}\right)>0$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part and $\nu$ : complex parameter
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E67
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

10.22.68 $\int_{0}^{\infty}t\exp\left(-p^{2}t^{2}\right)J_{0}\left(at\right)Y_{0}\left(% at\right)\,\mathrm{d}t=-\frac{1}{2\pi p^{2}}\exp\left(-\frac{a^{2}}{2p^{2}}% \right)K_{0}\left(\frac{a^{2}}{2p^{2}}\right),$ $\Re\left(p^{2}\right)>0$ .

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind and $\Re$ : real part
Referenced by:: §10.22(iv), §10.43(iv)
Permalink:: http://dlmf.nist.gov/10.22.E68
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E69
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

Equation (10.22.70) also remains valid if the order $\nu+1$ of the 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

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

In (10.22.71) and (10.22.72) are positive constants.

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $\nu$ : complex parameter
Keywords:: Mellin transform
Referenced by:: §10.22(iv), §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E71
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

ⓘ

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 , $\mathrm{e}$ : 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 , see §§14.3(i) and 14.3(ii), respectively.

In (10.22.74) and (10.22.75), are positive constants and

10.22.73

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

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\nu$ : complex parameter and : area
Keywords:: Mellin transform
Referenced by:: §10.22(iv), §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E74
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

If $|\nu|<\tfrac{1}{2}$ , then

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

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\nu$ : complex parameter and : area
Keywords:: Mellin transform
Referenced by:: §10.22(iv), §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.22.E75
Encodings:: TeX, pMML, png
See also:: Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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

§10.22(v) Hankel Transform

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

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

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

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

Hankel’s inversion theorem is given by

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

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

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

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

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

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

10.22.78 $f(x)=\int_{0}^{\infty}(xt)^{\frac{1}{2}}\frac{J_{\nu}\left(xt\right)Y_{\nu}% \left(at\right)-Y_{\nu}\left(xt\right)J_{\nu}\left(at\right)}{{J_{\nu}}^{2}% \left(at\right)+{Y_{\nu}}^{2}\left(at\right)}\*\int_{a}^{\infty}(yt)^{\frac{1}% {2}}\left(J_{\nu}\left(yt\right)Y_{\nu}\left(at\right)-Y_{\nu}\left(yt\right)J% _{\nu}\left(at\right)\right)f(y)\,\mathrm{d}y\,\mathrm{d}t,$

ⓘ

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

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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 , $\int$ : integral, : real variable, : real variable, $\nu$ : complex parameter and : function
Referenced by:: §1.18(ix), §1.18(vi), §10.22(v), §10.22(vi), Erratum (V1.2.0) for Chapter 10 Additions
Permalink:: http://dlmf.nist.gov/10.22.E79
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §10.22(v), §10.22 and Ch.10

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

§10.22(vi) Compendia

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

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