Bessel integral

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11: 10.71 Integrals

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${\displaystyle \int x{M}_{\nu }^2\left(x\right)dx}=x({\mathrm{ber}}_{\nu }x{\mathrm{bei}}_{\nu }^{\prime }x-{\mathrm{ber}}_{\nu }^{\prime }x{\mathrm{bei}}_{\nu }x),$

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${\displaystyle \int x{N}_{\nu }^2\left(x\right)dx}=x({\ker }_{\nu }x{\mathrm{kei}}_{\nu }^{\prime }x-{\ker }_{\nu }^{\prime }x{\mathrm{kei}}_{\nu }x),$

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12: 10.77 Software

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§10.77(ix) Integrals of Bessel Functions

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13: 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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14: Bibliography P

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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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Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

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External Links:

ISSN 0377-0427, Document, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

BibTeX

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A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

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A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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External Links:

ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

15: 10.54 Integral Representations

§10.54 Integral Representations

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10.54.1 $${𝗃}_n\left(z\right)=\frac{z^n}{2^{n+1}n!}{\int }_0^{\pi }\cos \left(z\cos \theta \right){(\sin \theta )}^{2n+1}d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.13

Referenced by:

§10.54

Permalink:

http://dlmf.nist.gov/10.54.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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10.54.2 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^n}{2}{\int }_0^{\pi }{\mathrm{e}}^{\mathrm{i}z\cos \theta }{P}_n\left(\cos \theta \right)\sin \theta d\theta .$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.14

Permalink:

http://dlmf.nist.gov/10.54.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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10.54.3 $${𝗄}_n\left(z\right)=\frac{\pi }{2}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{P}_n\left(t\right)dt,$$

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: 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, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

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${𝗁}_n^{(1)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

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16: 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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13.16.3 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\kappa -\frac{1}{2}}{J}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }\left(\kappa +\mu \right)+\frac{1}{2}>0$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.16.E3

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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17: 13.10 Integrals

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13.10.8 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{tz}{t}^{-a}𝐌(a,b,y/t)dt=\frac{1}{\mathrm{\Gamma }\left(a\right)}{z}^{\frac{1}{2}(2a-b-1)}{y}^{\frac{1}{2}(1-b)}{I}_{b-1}\left(2\sqrt{zy}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10(ii), §13.10 and Ch.13

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13.10.9 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{tz}{t}^{-a}U(a,b,y/t)dt=\frac{2z^{\frac{1}{2}(2a-b-1)}{y}^{\frac{1}{2}(1-b)}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}{K}_{b-1}\left(2\sqrt{zy}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $y$: real variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(ii), §13.10(ii), §13.10 and Ch.13

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13.10.13 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{b-1-\frac{1}{2}\nu }𝐌(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=x^{-a+\frac{1}{2}\nu }{\mathrm{e}}^{-x}𝐌(\nu -b+1,\nu -a+1,x),$$ $x>0$, , $\mathrm{\Re }b>0$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.10.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

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13.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}\nu }𝐌(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{x^{\frac{1}{2}\nu }{\mathrm{e}}^{-x}}{\mathrm{\Gamma }\left(b-a\right)}U(a,a-b+\nu +2,x),$$ $x>0$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric 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 $x$: real variable

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.10.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

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13.10.15 $${\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}\nu }U(a,b,t)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -b+2\right)}{\mathrm{\Gamma }\left(a\right)}{x}^{\frac{1}{2}\nu }U(\nu -b+2,\nu -a+2,x),$$ $x>0$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.10(v), §13.10 and Ch.13

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18: Bibliography G

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On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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External Links:

ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

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External Links:

ISSN 0771-050X, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:

ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.22(iv), §10.43(iv).

Encodings:

BibTeX

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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External Links:

ISSN 0044-2275, Document, MathReview

Referenced by:

§10.71(ii).

Encodings:

BibTeX

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19: 13.23 Integrals

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13.23.6 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{M}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{I}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.7 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{K}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.9 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\mu -\frac{1}{2}(\nu +1)}{M}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\kappa +\nu \right)}{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\kappa -\mu -\frac{3}{2})}{M}_{\frac{1}{2}(\kappa +3\mu -\nu +\frac{1}{2}),\frac{1}{2}(\kappa -\mu +\nu -\frac{1}{2})}\left(x\right),$$ $x>0$, ,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iii), §13.23 and Ch.13

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13.23.11 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{\frac{1}{2}t}{t}^{\frac{1}{2}(\nu -1)-\mu }{W}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -2\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}x}{x}^{\frac{1}{2}(\mu -\kappa -\frac{3}{2})}{W}_{\frac{1}{2}(\kappa +3\mu -\nu -\frac{1}{2}),\frac{1}{2}(\kappa -\mu +\nu +\frac{1}{2})}\left(x\right),$$ $x>0$, ,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.23.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iii), §13.23 and Ch.13

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13.23.12 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\frac{1}{2}(\nu -1)-\mu }{W}_{\kappa ,\mu }\left(t\right)J_{\nu }\left(2\sqrt{xt}\right)dt=\frac{\mathrm{\Gamma }\left(\nu -2\mu +1\right)}{\mathrm{\Gamma }\left(\frac{3}{2}-\mu -\kappa +\nu \right)}{\mathrm{e}}^{-\frac{1}{2}x}{x}^{\frac{1}{2}(\mu +\kappa -\frac{3}{2})}{M}_{\frac{1}{2}(\kappa -3\mu +\nu +\frac{1}{2}),\frac{1}{2}(\nu -\mu -\kappa +\frac{1}{2})}\left(x\right),$$ $x>0$, .

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(iii), §13.23 and Ch.13

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20: 8.6 Integral Representations

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8.6.2 $$\gamma (a,z)=z^{\frac{1}{2}a}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\frac{1}{2}a-1}{J}_a\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }a>0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: parameter

Referenced by:

§8.6(i)

Permalink:

http://dlmf.nist.gov/8.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(i), §8.6 and Ch.8

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8.6.6 $$\mathrm{\Gamma }(a,z)=\frac{2z^{\frac{1}{2}a}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(1-a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\frac{1}{2}a}{K}_a\left(2\sqrt{zt}\right)dt,$$ ,

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Symbols:

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

Referenced by:

§8.6(i)

Permalink:

http://dlmf.nist.gov/8.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(i), §8.6 and Ch.8

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