Bessel integral

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21: 11.5 Integral Representations

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11.5.7 $$I_{-\nu }\left(x\right)-{𝐋}_{\nu }\left(x\right)=\frac{2{(\frac{1}{2}x)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}{(1+t^2)}^{\nu -\frac{1}{2}}\sin \left(xt\right)dt,$$ $x>0$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable and $\nu$: real or complex order

Referenced by:

§11.5(i), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(i), §11.5 and Ch.11

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22: 13.4 Integral Representations

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13.4.2 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(b-c\right)}{\int }_0^1𝐌(a,c,zt)t^{c-1}{(1-t)}^{b-c-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }c>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $c$: arbitrary

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.3 $$𝐌(a,b,-z)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a-\frac{1}{2}b-\frac{1}{2}}{J}_{b-1}\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }a>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.6 $$U(a,b,z)=\frac{{(-1)}^n{z}^{1-b-n}}{\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{𝐌(b-a,b,t){\mathrm{e}}^{-t}{t}^{b+n-1}}{t+z}dt,$$ , $n=0,1,2,\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13.4.7 $$U(a,b,z)=\frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a-\frac{1}{2}b-\frac{1}{2}}{K}_{b-1}\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }a>\max (\mathrm{\Re }b-1,0)$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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23: 8.21 Generalized Sine and Cosine Integrals

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Spherical-Bessel-Function Expansions

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8.21.16 $$\mathrm{Si}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{3}{2}\right){\left(1-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}+\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k+1}\left(z\right),$$ $a\ne -1,-3,-5,\mathrm{\dots }$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Si}(a,z)$: generalized sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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8.21.17 $$\mathrm{Ci}(a,z)=z^a\sum _{k=0}^{\mathrm{\infty }}\frac{\left(2k+\frac{1}{2}\right){\left(\frac{1}{2}-\frac{1}{2}a\right)}_k}{{\left(\frac{1}{2}a\right)}_{k+1}}{𝗃}_{2k}\left(z\right),$$ $a\ne 0,-2,-4,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{Ci}(a,z)$: generalized cosine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.21(vi)

Permalink:

http://dlmf.nist.gov/8.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

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24: 10.75 Tables

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§10.75(iv) Integrals of Bessel Functions

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Abramowitz and Stegun (1964, Chapter 11) tabulates ${\int }_0^x{J}_0\left(t\right)dt$, ${\int }_0^x{Y}_0\left(t\right)dt$, $x=0(.1)10$, 10D; ${\int }_0^x{t}^{-1}(1-J_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}{Y}_0\left(t\right)dt$, $x=0(.1)5$, 8D.

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Zhang and Jin (1996, p. 270) tabulates ${\int }_0^x{J}_0\left(t\right)dt$, ${\int }_0^x{t}^{-1}(1-J_0\left(t\right))dt$, ${\int }_0^x{Y}_0\left(t\right)dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}{Y}_0\left(t\right)dt$, $x=0(.1)1(.5)20$, 8D.

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§10.75(v) Modified Bessel Functions and their Derivatives

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§10.75(vii) Integrals of Modified Bessel Functions

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25: 10.60 Sums

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10.60.11 $$\sum _{n=0}^{\mathrm{\infty }}{𝗃}_n^2\left(z\right)=\frac{\mathrm{Si}\left(2z\right)}{2z}.$$

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Symbols:

$\mathrm{Si}\left(z\right)$: sine integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.52

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

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26: Bibliography O

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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27: 9.11 Products

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§9.11(iii) Integral Representations

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9.11.3 $${\mathrm{Ai}}^2\left(x\right)=\frac{1}{4\pi \sqrt{3}}{\int }_0^{\mathrm{\infty }}{J}_0\left(\frac{1}{12}{t}^3+xt\right)tdt,$$ $x\ge 0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $x$: real variable

Source:

Lebedev (1965, Problem 22, p. 142)

Referenced by:

§9.5(i)

Permalink:

http://dlmf.nist.gov/9.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iii), §9.11 and Ch.9

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28: Bibliography L

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 217–231.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Integrals of Bessel Functions. McGraw-Hill Book Co., Inc., New York.

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External Links:

MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§10.22(ii), §10.22(iv), §10.22(vi), §10.43(iii), §10.43(vi), §11.7(v).

Encodings:

BibTeX

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29: 1.17 Integral and Series Representations of the Dirac Delta

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Bessel Functions and Spherical Bessel Functions (§§10.2(ii), 10.47(ii))

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1.17.13 $$\delta \left(x-a\right)=x{\int }_0^{\mathrm{\infty }}t{J}_{\nu }\left(xt\right)J_{\nu }\left(at\right)dt,$$ $\mathrm{\Re }\nu >-1$, $x>0$, $a>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Referenced by:

§1.18(vi)

Permalink:

http://dlmf.nist.gov/1.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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1.17.14 $$\delta \left(x-a\right)=\frac{2xa}{\pi }{\int }_0^{\mathrm{\infty }}{t}^2{𝗃}_{\mathrm{\ell }}\left(xt\right){𝗃}_{\mathrm{\ell }}\left(at\right)dt,$$ $x>0$, $a>0$.

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind

Referenced by:

§1.17(ii), §1.18(vi)

Permalink:

http://dlmf.nist.gov/1.17.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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30: 10.23 Sums

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10.23.19 $$a_m=\frac{2}{{(J_{\nu +1}\left(j_{\nu ,m}\right))}^2}{\int }_0^{1}tf(t)J_{\nu }\left(j_{\nu ,m}t\right)dt,$$ $\nu \ge -\frac{1}{2}$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\int$: integral, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $m$: integer, $\nu$: complex parameter, $a_k$ and $f(t)$

Referenced by:

§10.74(vii)

Permalink:

http://dlmf.nist.gov/10.23.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(iii), §10.23(iii), §10.23 and Ch.10

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