# Laplace transform

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##### 1: 1.14 Integral Transforms
###### Derivatives
1.14.31 $\mathscr{L}\left(f*g\right)=\mathscr{L}\left(f\right)\mathscr{L}\left(g\right).$
##### 2: 35.2 Laplace Transform
###### Convolution Theorem
If $g_{j}$ is the Laplace transform of $f_{j}$, $j=1,2$, then $g_{1}g_{2}$ is the Laplace transform of the convolution $f_{1}*f_{2}$, where …
##### 3: 16.20 Integrals and Series
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). …
##### 4: 19.13 Integrals of Elliptic Integrals
###### §19.13(iii) LaplaceTransforms
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.
##### 5: 2.5 Mellin Transform Methods
###### §2.5(iii) LaplaceTransforms with Small Parameters
2.5.37 $\mathscr{L}\mskip-3.0muh\mskip 3.0mu\left(\zeta\right)=\int_{0}^{\infty}h(t)e^% {-\zeta t}\,\mathrm{d}t.$
2.5.38 $\zeta\mathscr{L}\mskip-3.0muh\mskip 3.0mu\left(\zeta\right)=I_{1}(x)+I_{2}(x),$
2.5.45 $\mathscr{L}\mskip-3.0muh\mskip 3.0mu\left(\zeta\right)=\int_{0}^{\infty}\frac{% e^{-\zeta t}}{1+t}\,\mathrm{d}t,$ $\Re\zeta>0$.
2.5.49 $\mathscr{L}\mskip-3.0muh\mskip 3.0mu\left(\zeta\right)=e^{\zeta}E_{1}\left(% \zeta\right);$
##### 6: 15.14 Integrals
Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). Inverse Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §5.19), Oberhettinger and Badii (1973, §2.18), and Prudnikov et al. (1992b, §3.35). …
##### 8: 7.14 Integrals
###### LaplaceTransforms
7.14.2 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),$ $\Re a>0$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$,
7.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\sqrt{bt}\,\mathrm{d}t=\frac{1}{a}% \sqrt{\frac{b}{a+b}},$ $\Re a>0$, $\Re b>0$,
7.14.4 $\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t% }}\right)\,\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+% \sqrt{b})},$ $|\operatorname{ph}a|<\frac{1}{2}\pi$, $\Re b>0$, $\Re c\geq 0$.
##### 9: 3.11 Approximation Techniques
###### LaplaceTransform Inversion
Numerical inversion of the Laplace transform1.14(iii))
3.11.26 $F(s)=\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)=\int_{0}^{\infty}e^{-% st}f(t)\,\mathrm{d}t$
requires $f={\mathscr{L}}^{-1}F$ to be obtained from numerical values of $F$. A general procedure is to approximate $F$ by a rational function $R$ (vanishing at infinity) and then approximate $f$ by $r={\mathscr{L}}^{-1}R$. …
##### 10: 2.4 Contour Integrals
Then … For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985).
###### §2.4(ii) Inverse LaplaceTransforms
Then the Laplace transformFor examples see Olver (1997b, pp. 315–320). …