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Appendix B Table of Laplace Transforms

The function \(u\) is the Heaviside function, \(\delta\) is the Dirac delta function, and

\begin{equation*} \Gamma(t) = \int_0^\infty e^{-\tau} \tau^{t-1} \, d\tau , \qquad \operatorname{erf}(t) = \frac{2}{\sqrt{\pi}} \int_0^t e^{-\tau^2} \, d\tau , \qquad \operatorname{erfc}(t) = 1 - \operatorname{erf}(t) . \end{equation*}

\(f(t)\)   \(F(s) = \mathcal{L} \bigl\{ f(t) \bigr\}= \int_0^\infty e^{-st} f(t) \, dt\)
\(C\)   \(\frac{C}{s}\)
\(t\)   \(\frac{1}{s^2}\)
\(t^2\)   \(\frac{2}{s^3}\)
\(t^n\)   \(\frac{n!}{s^{n+1}}\)
\(t^p \quad (p > 0)\)   \(\frac{\Gamma(p+1)}{s^{p+1}}\)
\(e^{-at}\)   \(\frac{1}{s+a}\)
\(\sin (\omega t)\)   \(\frac{\omega}{s^2+\omega^2}\)
\(\cos (\omega t)\)   \(\frac{s}{s^2+\omega^2}\)
\(\sinh (\omega t)\)   \(\frac{\omega}{s^2-\omega^2}\)
\(\cosh (\omega t)\)   \(\frac{s}{s^2-\omega^2}\)
\(u(t-a)\)   \(\frac{e^{-as}}{s}\)
\(\delta(t)\)   \(1\)
\(\delta(t-a)\)   \(e^{-as}\)
\(\operatorname{erf}\left( \frac{t}{2a} \right)\)   \(\frac{1}{s} e^{(as)^2} \operatorname{erfc}(as)\)
\(\frac{1}{\sqrt{\pi t}} \exp\left(\frac{-a^2}{4t}\right) \quad (a \geq 0)\)   \(\frac{e^{-as}}{\sqrt{s}}\)
\(\frac{1}{\sqrt{\pi t}} - a e^{a^2 t} \operatorname{erfc}(a \sqrt{t}) \quad (a>0)\)   \(\frac{1}{\sqrt{s}+a}\)
\(a f(t) + b g(t)\)   \(a F(s) + bG(s)\)
\(f(at) \quad (a > 0)\)   \(\frac{1}{a}F\left( \frac{s}{a} \right)\)
\(f(t-a)u(t-a)\)   \(e^{-as} F(s)\)
\(e^{-at} f(t)\)   \(F(s+a)\)
\(g'(t)\)   \(sG(s)-g(0)\)
\(g''(t)\)   \(s^2G(s)-sg(0)-g'(0)\)
\(g'''(t)\)   \(s^3G(s)-s^2g(0)-sg'(0)-g''(0)\)
\(g^{(n)}(t)\)   \(s^nG(s)-s^{n-1}g(0)-\cdots-g^{(n-1)}(0)\)
\((f * g)(t) = \int_0^t f(\tau) g(t-\tau) \, d\tau\)   \(F(s)G(s)\)
\(tf(t)\)   \(-F'(s)\)
\(t^nf(t)\)   \({(-1)}^nF^{(n)}(s)\)
\(\int_0^t f(\tau) d\tau\)   \(\frac{1}{s} F(s)\)
\(\frac{f(t)}{t}\)   \(\int_s^\infty F(\sigma) d\sigma\)

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