integrals of modified Bessel functions

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11: 13.16 Integral Representations

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13.16.2 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\lambda }}{\mathrm{\Gamma }\left(1+2\mu -2\lambda \right)\mathrm{\Gamma }\left(2\lambda \right)}{\int }_0^1{M}_{\kappa -\lambda ,\mu -\lambda }\left(zt\right){\mathrm{e}}^{\frac{1}{2}z(t-1)}{t}^{\mu -\lambda -\frac{1}{2}}{(1-t)}^{2\lambda -1}dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\mathrm{\Re }\lambda >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker 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:

§13.23(v)

Permalink:

http://dlmf.nist.gov/13.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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13.16.4 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{-\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\kappa -\frac{1}{2}}{I}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

Erratum (V1.0.5) for Equation (13.16.4)

Permalink:

http://dlmf.nist.gov/13.16.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the condition for the validity of this equation was stated incorrectly as $\mathrm{\Re }(\kappa -\mu )-\frac{1}{2}>0$. The correct condition is .

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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13.16.8 $$W_{\kappa ,\mu }\left(z\right)=\frac{2\sqrt{z}{\mathrm{e}}^{-\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\kappa -\frac{1}{2}}{K}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }\left(\mu -\kappa \right)+\frac{1}{2}>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker 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/13.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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12: Bibliography P

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

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Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.

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External Links:

ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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13: 13.4 Integral Representations

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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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14: 9.12 Scorer Functions

… ►If $\zeta =\frac{2}{3}{z}^{3/2}$ or $\frac{2}{3}{x}^{3/2}$, and $K_{1/3}$ is the modified Bessel function (§10.25(ii)), then ►

9.12.22 $$\mathrm{Hi}\left(-z\right)=\frac{4z^2}{3^{3/2}{\pi }^2}{\int }_0^{\mathrm{\infty }}\frac{K_{1/3}\left(t\right)}{{\zeta }^2+t^2}dt,$$ ,

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Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\zeta$: variable

Source:

Gordon (1970, Appendix A)

Permalink:

http://dlmf.nist.gov/9.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

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9.12.23 $$\mathrm{Gi}\left(x\right)=\frac{4x^2}{3^{3/2}{\pi }^2}{⨍}_0^{\mathrm{\infty }}\frac{K_{1/3}\left(t\right)}{{\zeta }^2-t^2}dt,$$ $x>0$,

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${⨍}_a^b$: Cauchy principal value, $x$: real variable and $\zeta$: variable

Source:

Gordon (1970, Appendix A)

Permalink:

http://dlmf.nist.gov/9.12.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

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15: 10.22 Integrals

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10.22.52 $${\int }_0^{\mathrm{\infty }}{J}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt=\frac{\sqrt{\pi }}{2p}\exp \left(-\frac{b^2}{8p^2}\right)I_{\nu /2}\left(\frac{b^2}{8p^2}\right),$$ $\mathrm{\Re }\nu >-1,\mathrm{\Re }\left(p^2\right)>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.22.E52

Encodings:

pMML, png, TeX

See also:

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

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10.22.53 $${\int }_0^{\mathrm{\infty }}{Y}_{2\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt=-\frac{\sqrt{\pi }}{2p}\exp \left(-\frac{b^2}{8p^2}\right)\left(I_{\nu }\left(\frac{b^2}{8p^2}\right)\tan \left(\nu \pi \right)+\frac{1}{\pi }{K}_{\nu }\left(\frac{b^2}{8p^2}\right)\sec \left(\nu \pi \right)\right),$$ , $\mathrm{\Re }\left(p^2\right)>0$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function, $\tan z$: tangent function and $\nu$: complex parameter

A&S Ref:

11.4.30

Permalink:

http://dlmf.nist.gov/10.22.E53

Encodings:

pMML, png, TeX

See also:

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

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10.22.67 $${\int }_0^{\mathrm{\infty }}t\exp \left(-p^2{t}^2\right)J_{\nu }\left(at\right)J_{\nu }\left(bt\right)dt=\frac{1}{2p^2}\exp \left(-\frac{a^2+b^2}{4p^2}\right)I_{\nu }\left(\frac{ab}{2p^2}\right),$$ $\mathrm{\Re }\nu >-1,\mathrm{\Re }\left(p^2\right)>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E67

Encodings:

pMML, png, TeX

See also:

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

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10.22.68 $${\int }_0^{\mathrm{\infty }}t\exp \left(-p^2{t}^2\right)J_0\left(at\right)Y_0\left(at\right)dt=-\frac{1}{2\pi p^2}\exp \left(-\frac{a^2}{2p^2}\right)K_0\left(\frac{a^2}{2p^2}\right),$$ $\mathrm{\Re }\left(p^2\right)>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $\mathrm{\Re }$: real part

Referenced by:

§10.22(iv), §10.43(iv)

Permalink:

http://dlmf.nist.gov/10.22.E68

Encodings:

pMML, png, TeX

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). …

16: Bibliography G

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Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. 420 (1), pp. 373–386.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Nihat Yagmur), ZentralBlatt

Referenced by:

§10.37, §10.37.

Encodings:

BibTeX

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Some integrals involving three modified Bessel functions. I. J. Math. Phys. 27 (3), pp. 682–687.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Some integrals involving three modified Bessel functions. II. J. Math. Phys. 27 (3), pp. 688–695.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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17: 6.7 Integral Representations

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6.7.15 $$\mathrm{f}\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_0\left(2\sqrt{zt}\right)\cos tdt,$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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6.7.16 $$\mathrm{g}\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_0\left(2\sqrt{zt}\right)\sin tdt.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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18: 10.9 Integral Representations

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10.9.28 $$J_{\nu }\left(z\right)J_{\nu }\left(\zeta \right)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\exp \left(\frac{1}{2}t-\frac{z^2+{\zeta }^2}{2t}\right)I_{\nu }\left(\frac{z\zeta }{t}\right)\frac{dt}{t},$$ $\mathrm{\Re }\nu >-1$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Permalink:

http://dlmf.nist.gov/10.9.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(iii), §10.9 and Ch.10

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10.9.30 $$J_{\nu }^2\left(z\right)+Y_{\nu }^2\left(z\right)=\frac{8}{{\pi }^2}{\int }_0^{\mathrm{\infty }}\cosh \left(2\nu t\right)K_0\left(2z\sinh t\right)dt,$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.9.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

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19: 10.21 Zeros

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10.21.17 $$\frac{dc}{d\nu }=2c{\int }_0^{\mathrm{\infty }}{K}_0\left(2c\sinh t\right){\mathrm{e}}^{-2\nu t}dt,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\nu$: complex parameter and $c$: zero of cylinder function

Permalink:

http://dlmf.nist.gov/10.21.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(iv), §10.21 and Ch.10

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10.21.18 $$\frac{d{c}^{\prime }}{d\nu }=\frac{2c^{\prime }}{c_{}^{\prime }{}_{}{}^2-{\nu }^2}{\int }_0^{\mathrm{\infty }}(c_{}^{\prime }{}_{}{}^2\cosh \left(2t\right)-{\nu }^2)K_0\left(2c^{\prime }\sinh t\right){\mathrm{e}}^{-2\nu t}dt,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\nu$: complex parameter and $c$: zero of cylinder function

Permalink:

http://dlmf.nist.gov/10.21.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(iv), §10.21 and Ch.10

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20: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.23 $$\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }+2}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+2)\varphi \right){\mathrm{se}}_{2m+2}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+2}^{2m+2}(h^2){\mathrm{Ms}}_{2m+2}^{(j)}(z,h).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.28 (in different form and only for $\mathrm{\ell }=0$)

Permalink:

http://dlmf.nist.gov/28.28.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

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