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*}
| \(f(t)\) | \(F(s) = \mathcal{L} \bigl\{ f(t) \bigr\}= \int_0^\infty e^{-st} f(t) \, dt\) |
| \(C\) | \(\frac{C}{s}\) |
| \(t\) | \(\frac{1}{s^2}\) |
| \(t^n\) | \(\frac{n!}{s^{n+1}}\) |
| \(t^p \quad (p > -1)\) | \(\frac{\Gamma(p+1)}{s^{p+1}}\) |
| \(e^{-at}\) | \(\frac{1}{s+a}\) |
| \(\sin (\omega t)\) | \(\frac{\omega}{s^2+\omega^2}\) |
| \(\cos (\omega t)\) | \(\frac{s}{s^2+\omega^2}\) |
| \(\sinh (\omega t)\) | \(\frac{\omega}{s^2-\omega^2}\) |
| \(\cosh (\omega t)\) | \(\frac{s}{s^2-\omega^2}\) |
| \(u(t-a) \quad (a \geq 0)\) | \(\frac{e^{-as}}{s}\) |
| \(\delta(t)\) | \(1\) |
| \(\delta(t-a) \quad (a \geq 0)\) | \(e^{-as}\) |
| \(\operatorname{erf}\left( \frac{t}{2a} \right)\) | \(\frac{1}{s} e^{(as)^2} \operatorname{erfc}(as)\) |
| \(t \sin(\omega t)\) | \(\frac{2\omega s}{{(s^2+\omega^2)}^2}\) |
| \(t \cos(\omega t)\) | \(\frac{s^2-\omega^2}{{(s^2+\omega^2)}^2}\) |
| \(e^{-at} \sin(\omega t)\) | \(\frac{\omega}{{(s+a)}^2 + \omega^2}\) |
| \(e^{-at} \cos(\omega t)\) | \(\frac{s+a}{{(s+a)}^2 + \omega^2}\) |
| \(e^{-at} \sinh(\omega t)\) | \(\frac{\omega}{{(s+a)}^2 - \omega^2}\) |
| \(e^{-at} \cosh(\omega t)\) | \(\frac{s+a}{{(s+a)}^2 - \omega^2}\) |
| \(\frac{1}{\sqrt{\pi t}} \exp\left(\frac{-a^2}{4t}\right) \quad (a > 0)\) | \(\frac{e^{-a\sqrt{s}}}{\sqrt{s}}\) |
| \(\frac{a}{\sqrt{4\pi t^3}} \exp\left(\frac{-a^2}{4t}\right) \quad (a > 0)\) | \(e^{-a \sqrt{s}}\) |
| \(\frac{1}{\sqrt{\pi t}} - a e^{a^2 t} \operatorname{erfc}(a \sqrt{t}) \quad (a>0)\) | \(\frac{1}{\sqrt{s}+a}\) |
| \(\operatorname{erfc}\left(\frac{a}{2\sqrt{t}}\right) \quad (a>0)\) | \(\frac{e^{-a\sqrt{s}}}{s}\) |
| \(\frac{1}{2\sqrt{\pi t^3}}( e^{b t} - e^{a t})\) | \(\sqrt{s-a}-\sqrt{s-b}\) |
| \(\frac{1}{t} ( e^{b t} - e^{a t})\) | \(\ln \frac{s-a}{s-b}\) |
| \(J_0(at)\) | \(\frac{1}{\sqrt{s^2+a^2}}\) |
| \(J_0(a\sqrt{t})\) | \(\frac{\exp\left(\frac{-a^2}{4s}\right)}{s}\) |
| \(\frac{1}{\sqrt{\pi t}} \cos(a \sqrt{t})\) | \(\frac{\exp\left(\frac{-a^2}{4s}\right)}{\sqrt{s}}\) |
| \(\frac{2}{\sqrt{\pi}} \sinh(a \sqrt{t})\) | \(\frac{a \exp\left(\frac{a^2}{4s}\right)}{s^{3/2}}\) |
| \(\frac{1}{\sqrt{\pi t}} e^{at} (1+2at)\) | \(\frac{s}{(s-a)^{3/2}}\) |
| \(a f(t) + b g(t)\) | \(a F(s) + bG(s)\) |
| \(f(at) \quad (a > 0)\) | \(\frac{1}{a}F\left( \frac{s}{a} \right)\) |
| \(f(t-a)u(t-a) \quad (a \geq 0)\) | \(e^{-as} F(s)\) |
| \(e^{-at} f(t)\) | \(F(s+a)\) |
| \(g'(t)\) | \(sG(s)-g(0)\) |
| \(g''(t)\) | \(s^2G(s)-sg(0)-g'(0)\) |
| \(g^{(n)}(t)\) | \(s^nG(s)-s^{n-1}g(0)-\cdots-g^{(n-1)}(0)\) |
| \((f * g)(t) = \int_0^t f(\tau) g(t-\tau) \, d\tau\) | \(F(s)G(s)\) |
| \(tf(t)\) | \(-F'(s)\) |
| \(t^nf(t)\) | \({(-1)}^nF^{(n)}(s)\) |
| \(\int_0^t f(\tau) d\tau\) | \(\frac{1}{s} F(s)\) |
| \(\frac{f(t)}{t}\) | \(\int_s^\infty F(\sigma) d\sigma\) |
| \(f(t)\) periodic with period \(P\) | \(\frac{1}{1-e^{-Ps}} \int_0^P e^{-st} f(t) \, dt\) |
