Laplace integral

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31: 16.20 Integrals and Series

§16.20 Integrals and Series

►Integrals of the Meijer $G$-function are given in Apelblat (1983, §19), Erdélyi et al. (1953a, §5.5.2), Erdélyi et al. (1954a, §§6.9 and 7.5), Luke (1969a, §3.6), Luke (1975, §5.6), Mathai (1993, §3.10), and Prudnikov et al. (1990, §2.24). Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer $G$-function are given in Prudnikov et al. (1992a, §3.40) and Prudnikov et al. (1992b, §3.38). …

32: 9.10 Integrals

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9.10.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{-p^3/3}\left(\frac{1}{3}-\frac{p{}_1{}^{}F_1^{}(\frac{1}{3};\frac{4}{3};\frac{1}{3}{p}^3)}{3^{4/3}\mathrm{\Gamma }\left(\frac{4}{3}\right)}+\frac{p^2{}_1{}^{}F_1^{}(\frac{2}{3};\frac{5}{3};\frac{1}{3}{p}^3)}{3^{5/3}\mathrm{\Gamma }\left(\frac{5}{3}\right)}\right),$$ $p\in ℂ$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E14

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See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(-t\right)dt=\frac{1}{3}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}+\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right),$$ $\mathrm{\Re }p>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E15

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See also:

Annotations for §9.10(v), §9.10 and Ch.9

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9.10.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Bi}\left(-t\right)dt=\frac{1}{\sqrt{3}}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}-\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$ $\mathrm{\Re }p>0$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter

Keywords:

Laplace transform

Source:

Gibbs (1973, Problem 72–21)

Permalink:

http://dlmf.nist.gov/9.10.E16

Encodings:

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See also:

Annotations for §9.10(v), §9.10 and Ch.9

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33: 14.31 Other Applications

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§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

34: Bibliography H

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Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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External Links:

ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§17.13.

Encodings:

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

Encodings:

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Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

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External Links:

MathReview (C. Tanaka)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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Elliptic Integrals. Dover Publications Inc., New York.

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Notes:

Unaltered reprint of original edition published by Wiley, New York, 1917.

External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§19.8(ii).

Encodings:

BibTeX

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Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.

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External Links:

Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(iii), §2.5(ii).

Encodings:

BibTeX

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35: 10.32 Integral Representations

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10.32.16 $$I_{\mu }\left(x\right)K_{\nu }\left(x\right)={\int }_0^{\mathrm{\infty }}{J}_{\mu \pm \nu }\left(2x\sinh t\right){\mathrm{e}}^{(-\mu \pm \nu )t}dt,$$ $\mathrm{\Re }\left(\mu \mp \nu \right)>-\frac{1}{2}$, $\mathrm{\Re }\left(\mu \pm \nu \right)>-1$, $x>0$.

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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, $\sinh z$: hyperbolic sine 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, $x$: real variable and $\nu$: complex parameter

Keywords:

Laplace transform

Referenced by:

§10.32(iii)

Permalink:

http://dlmf.nist.gov/10.32.E16

Encodings:

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See also:

Annotations for §10.32(iii), §10.32 and Ch.10

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36: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

Encodings:

BibTeX

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The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt

Referenced by:

§1.14(iii), §1.14(vii).

Encodings:

BibTeX

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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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External Links:

Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:

ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

Encodings:

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Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:

Double-precision Fortran.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

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37: 10.22 Integrals

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10.22.49 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{J}_{\nu }\left(bt\right)dt=\frac{{(\frac{1}{2}b)}^{\nu }}{a^{\mu +\nu }}\mathrm{\Gamma }\left(\mu +\nu \right)𝐅(\frac{\mu +\nu }{2},\frac{\mu +\nu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\mu +\nu \right)>0,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ 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:

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See also:

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

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10.22.66 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{J}_{\nu }\left(bt\right)J_{\nu }\left(ct\right)dt=\frac{1}{\pi {(bc)}^{\frac{1}{2}}}{Q}_{\nu -\frac{1}{2}}\left(\frac{a^2+b^2+c^2}{2bc}\right),$$ $\mathrm{\Re }\nu >-\frac{1}{2}$.

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: 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:

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See also:

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

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38: 20.10 Integrals

§20.10 Integrals

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

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See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

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See also:

Annotations for §20.10(ii), §20.10 and Ch.20

… ►For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).

39: 10.43 Integrals

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10.43.22

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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 $\nu$: complex parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E22

Encodings:

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See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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40: Bibliography N

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.16), (11.11.17), §11.11(iii).

Encodings:

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Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:

Includes Algol program.

External Links:

MathReview

Referenced by:

§19.39(iii).

Encodings:

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On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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