Laplace transforms

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21: 13.23 Integrals

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§13.23(i) Laplace and Mellin Transforms

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13.23.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)}{{\left(z+\frac{1}{2}\right)}^{\mu +\nu +\frac{1}{2}}}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}+\mu +\nu }{1+2\mu };\frac{1}{z+\frac{1}{2}}),$$ $\mathrm{\Re }\mu +\nu +\frac{1}{2}>0$, $\mathrm{\Re }z>\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric 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

Keywords:

Laplacetransform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E1

Encodings:

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See also:

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

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13.23.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(2\mu +1\right){\left(z+\frac{1}{2}\right)}^{-\kappa -\mu -\frac{1}{2}}{\left(z-\frac{1}{2}\right)}^{\kappa -\mu -\frac{1}{2}},$$ $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }z>\frac{1}{2}$,

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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

Keywords:

Laplacetransform, Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E2

Encodings:

pMML, png, TeX

See also:

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

… ►For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), and Prudnikov et al. (1992a, §§3.34, 3.35). Inverse Laplace transforms are given in Oberhettinger and Badii (1973, §2.16) and Prudnikov et al. (1992b, §§3.33, 3.34). …

22: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: …

23: 18.17 Integrals

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§18.17(vi) Laplace Transforms

►Many of the Fourier transforms given in §18.17(v) have analytic continuations to Laplace transforms. … ►

Jacobi

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Laguerre

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Hermite

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24: Bibliography P

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Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:

Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.

External Links:

ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).

Encodings:

BibTeX

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Integrals and Series: Inverse Laplace Transforms, Vol. 5. Gordon and Breach Science Publishers, New York.

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External Links:

ISBN 2-88124-838-1, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §13.23(iv), §14.17(v), §15.14, §15.14, §16.15, §16.20, §16.5, §18.17(ix), §19.13(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.6(iii), §9.10(ix).

Encodings:

BibTeX

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25: 7.7 Integral Representations

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7.7.4 $${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{\sqrt{t+z^2}}dt=\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{a{z}^2}\mathrm{erfc}\left(\sqrt{a}z\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplacetransform

A&S Ref:

7.4.8

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E4

Encodings:

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See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\cos \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: 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, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplacetransform

A&S Ref:

7.4.22 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E15

Encodings:

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See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\sin \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{g}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: 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 and $\sin z$: sine function

Keywords:

Laplacetransform

A&S Ref:

7.4.23 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E16

Encodings:

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See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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26: 3.5 Quadrature

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Example. Laplace Transform Inversion

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3.5.38 $$G(p)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}g(t)dt,$$

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Defines:

$G(p)$: Laplace transform of $g(t)$ (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $g(t)$: function

A&S Ref:

29.2.1 (in different form)

Permalink:

http://dlmf.nist.gov/3.5.E38

Encodings:

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See also:

Annotations for §3.5(viii), §3.5(viii), §3.5 and Ch.3

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Table 3.5.19: Laplace transform inversion.

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$t$	$J_0(t)$	$g(t)$
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… ►In fact from (7.14.4) and the inversion formula for the Laplace transform (§1.14(iii)) we have … ►A special case is the rule for Hilbert transforms (§1.14(v)): …

27: Bibliography H

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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:

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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28: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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Laplace Transform

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29: 8.19 Generalized Exponential Integral

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8.19.2 $$E_p\left(z\right)=z^{p-1}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t^p}dt.$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $z$: complex variable and $p$: parameter

Keywords:

Laplacetransform, Mellin transform

Referenced by:

§8.19(i), §8.19(viii)

Permalink:

http://dlmf.nist.gov/8.19.E2

Encodings:

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See also:

Annotations for §8.19(i), §8.19 and Ch.8

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8.19.3 $$E_p\left(z\right)={\int }_1^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^p}dt,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $p$: parameter

Keywords:

Laplacetransform, Mellin transform

A&S Ref:

5.1.4

Referenced by:

§8.19(iii)

Permalink:

http://dlmf.nist.gov/8.19.E3

Encodings:

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See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

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8.19.4 $$E_p\left(z\right)=\frac{z^{p-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{p-1}{\mathrm{e}}^{-zt}}{1+t}dt,$$ , $\mathrm{\Re }p>0$.

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $p$: parameter

Keywords:

Laplacetransform, Mellin transform

Permalink:

http://dlmf.nist.gov/8.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(i), §8.19(i), §8.19 and Ch.8

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30: Bibliography D

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

Encodings:

BibTeX

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