12: Appendix B- Table of Laplace Transforms
The function \(u\) is the Heaviside function, \(\delta\) is the Dirac delta function, and
\[\Gamma(t)=\int_{0}^{\infty}e^{-\tau}\tau^{t-1}d\tau ,\qquad\text{erf}(t)=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-\tau^{2}}d\tau , \qquad\text{erfc}(t)=1-\text{erf}(t).\]
Table \(\PageIndex{1}\)
| \(f(t)\) | \(F(s)=\mathcal{L}\{f(t)\}=\int_{0}^{\infty}e^{-st}f(t)dt\) |
|---|---|
| \(C\) | \(\frac{C}{s}\) |
| \(t\) | \(\frac{1}{s^{2}}\) |
| \(t^{2}\) | \(\frac{2}{s^{3}}\) |
| \(t^{n}\) | \(\frac{n!}{s^{n+1}}\) |
| \(t^{p}\quad (p>0)\) | \(\frac{\Gamma (p+1)}{s^{p+1}}\) |
| \(e^{-at}\) | \(\frac{1}{s+a}\) |
| \(\sin(\omega t)\) | \(\frac{\omega}{s^{2}+\omega ^{2}}\) |
| \(\cos(\omega t)\) | \(\frac{s}{s^{2}+\omega^{2}}\) |
| \(\sinh (\omega t)\) | \(\frac{\omega}{s^{2}-\omega ^{2}}\) |
| \(\cosh (\omega t)\) | \(\frac{s}{s^{2}-\omega^{2}}\) |
| \(u(t-a)\) | \(\frac{e^{-as}}{s}\) |
| \(\delta (t)\) | \(1\) |
| \(\delta (t-a)\) | \(e^{-as}\) |
| \(\text{erf}\left(\frac{t}{2a}\right)\) | \(\frac{1}{s}e^{(as)^{2}}\text{erfc}(as)\) |
| \(\frac{1}{\sqrt{\pi t}}\text{exp}\left(\frac{-a^{2}}{4t}\right)\quad (a\geq 0)\) | \(\frac{e^{-as}}{\sqrt{s}}\) |
| \(\frac{1}{\sqrt{\pi t}}-ae^{a^{2}t}\text{erfc}(a\sqrt{t})\quad (a>0)\) | \(\frac{1}{\sqrt{s}+a}\) |
| \(af(t)+bg(t)\) | \(aF(s)+bG(s)\) |
| \(f(at)\quad (a>0)\) | \(\frac{1}{a}F\left(\frac{s}{a}\right)\) |
| \(f(t-a)u(t-a)\) | \(e^{-as}F(s)\) |
| \(e^{-at}f(t)\) | \(F(s+a)\) |
| \(g'(t)\) | \(sG(s)-g(0)\) |
| \(g''(t)\) | \(s^{2}G(s)-sg(0)-g'(0)\) |
| \(g'''(t)\) | \(s^{3}G(s)-s^{2}g(0)-sg'(0)-g''(0)\) |
| \(g^{(n)}(t)\) | \(s^{n}G(s)-s^{n-1}g(0)-\cdots -g^{(n-1)}(0)\) |
| \((f\ast g)(t)=\int_{0}^{t} f(\tau )g(t-\tau )d\tau \) | \(F(s)G(s)\) |
| \(tf(t)\) | \(-F'(s)\) |
| \(t^{n}f(t)\) | \((-1)^{n}F^{(n)}(s)\) |
| \(\int_{0}^{t}f(\tau )d\tau \) | \(\frac{1}{s}F(s)\) |
| \(\frac{f(t)}{t}\) | \(\int_{s}^{\infty} F(\sigma )d\sigma\) |