Laplace integral

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1: 6.14 Integrals

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§6.14(i) Laplace Transforms

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2: 19.13 Integrals of Elliptic Integrals

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§19.13(iii) Laplace Transforms

►For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.

3: 7.14 Integrals

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Laplace Transforms

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Laplace Transforms

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}+a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$,

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.29 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}-a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$.

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Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.30 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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4: 10.71 Integrals

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5: 2.3 Integrals of a Real Variable

… ►Assume that the Laplace transform ►

2.3.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt$$

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Symbols:

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

Keywords:

Laplace transform

Referenced by:

§2.3(ii)

Permalink:

http://dlmf.nist.gov/2.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(i), §2.3 and Ch.2

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2.3.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\frac{q^{(s)}(0)}{x^{s+1}},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function

Keywords:

Laplace transform

Referenced by:

§2.3(i)

Permalink:

http://dlmf.nist.gov/2.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(i), §2.3 and Ch.2

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§2.3(iii) Laplace’s Method

… ►These references and Wong (1989, Chapter 2) also contain examples. …

6: 16.5 Integral Representations and Integrals

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16.5.3 $${}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{1}{\mathrm{\Gamma }\left(a_0\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a_0-1}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\cdots },b_q};zt)dt,$$ , $\mathrm{\Re }{a}_0>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Keywords:

Laplace transform, Mellin transform

Referenced by:

§16.5

Permalink:

http://dlmf.nist.gov/16.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

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7: 2.4 Contour Integrals

… ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). … ►

2.4.2 $$Q(z)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}q(t)dt$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $q(t)$: analytic function and $Q(z)$: Laplace transform of $q(t)$

Permalink:

http://dlmf.nist.gov/2.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(ii), §2.4 and Ch.2

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2.4.5 $$q(t)=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{tz}Q(z)dz,$$ ,

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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, $\mathrm{i}$: imaginary unit, $\int$: integral, $q(t)$: analytic function, $Q(z)$: Laplace transform of $q(t)$ and $\sigma$: constant

Permalink:

http://dlmf.nist.gov/2.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(ii), §2.4 and Ch.2

… ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

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8: 35.4 Partitions and Zonal Polynomials

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Laplace and Beta Integrals

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9: 14.17 Integrals

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§14.17(v) Laplace Transforms

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10: 19.18 Derivatives and Differential Equations

… ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). … ►and $U=R_{-a}(\frac{1}{2},\frac{1}{2};z+\mathrm{i}\rho ,z-\mathrm{i}\rho )$, with $\rho =\sqrt{x^2+y^2}$, satisfies Laplace’s equation: …