Laplace integral

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2: 19.13 Integrals of Elliptic Integrals
§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
Laplace Transforms
7.14.7 $\int_{0}^{\infty}e^{-at}C\left(\sqrt{\frac{2t}{\pi}}\right)\mathrm{d}t=\frac{(% \sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},$ $\Re a>0$,
7.14.8 $\int_{0}^{\infty}e^{-at}S\left(\sqrt{\frac{2t}{\pi}}\right)\mathrm{d}t=\frac{(% \sqrt{a^{2}+1}-a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},$ $\Re a>0$.
5: 2.3 Integrals of a Real Variable
Assume that the Laplace transform
2.3.2 $\int_{0}^{\infty}e^{-xt}q(t)\mathrm{d}t\sim\sum_{s=0}^{\infty}\frac{q^{(s)}(0)% }{x^{s+1}},$ $x\to+\infty$.
§2.3(iii) Laplace’s Method
These references and Wong (1989, Chapter 2) also contain examples. …
6: 16.5 Integral Representations and Integrals
16.5.3 ${{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\mathrm{d}t,$ $\Re z<1$, $\Re a_{0}>0$,
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 $Q(z)=\int_{0}^{\infty}e^{-zt}q(t)\mathrm{d}t$
2.4.5 $q(t)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}e^{tz}Q(z)\mathrm{d% }z,$ $0,
For examples see Olver (1997b, pp. 315–320).
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 $R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)$ (see (19.16.20) and (19.16.23)). … and $U=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z+i\rho,z-i\rho\right)$, with $\rho=\sqrt{x^{2}+y^{2}}$, satisfies Laplace’s equation: …