Appendix B: Table of Laplace Transforms
- Page ID
- 73041
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The function \(u\) is the Heaviside function, \(\delta\) is the Dirac delta function, and
\[\Gamma(t)=\int_{0}^{\infty}e^{-\tau}\tau^{t-1}d\tau ,\qquad\text{erf}(t)=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-\tau^{2}}d\tau , \qquad\text{erfc}(t)=1-\text{erf}(t). \nonumber \]
Table \(\PageIndex{1}\)
| \(f(t)\) | \(F(s)=\mathcal{L}\{f(t)\}=\int_{0}^{\infty}e^{-st}f(t)dt\) |
|---|---|
| \(C\) | \(\frac{C}{s}\) |
| \(t\) | \(\frac{1}{s^{2}}\) |
| \(t^{2}\) | \(\frac{2}{s^{3}}\) |
| \(t^{n}\) | \(\frac{n!}{s^{n+1}}\) |
| \(t^{p}\quad (p>0)\) | \(\frac{\Gamma (p+1)}{s^{p+1}}\) |
| \(e^{-at}\) | \(\frac{1}{s+a}\) |
| \(\sin(\omega t)\) | \(\frac{\omega}{s^{2}+\omega ^{2}}\) |
| \(\cos(\omega t)\) | \(\frac{s}{s^{2}+\omega^{2}}\) |
| \(\sinh (\omega t)\) | \(\frac{\omega}{s^{2}-\omega ^{2}}\) |
| \(\cosh (\omega t)\) | \(\frac{s}{s^{2}-\omega^{2}}\) |
| \(u(t-a)\) | \(\frac{e^{-as}}{s}\) |
| \(\delta (t)\) | \(1\) |
| \(\delta (t-a)\) | \(e^{-as}\) |
| \(\text{erf}\left(\frac{t}{2a}\right)\) | \(\frac{1}{s}e^{(as)^{2}}\text{erfc}(as)\) |
| \(\frac{1}{\sqrt{\pi t}}\text{exp}\left(\frac{-a^{2}}{4t}\right)\quad (a\geq 0)\) | \(\frac{e^{-as}}{\sqrt{s}}\) |
| \(\frac{1}{\sqrt{\pi t}}-ae^{a^{2}t}\text{erfc}(a\sqrt{t})\quad (a>0)\) | \(\frac{1}{\sqrt{s}+a}\) |
| \(af(t)+bg(t)\) | \(aF(s)+bG(s)\) |
| \(f(at)\quad (a>0)\) | \(\frac{1}{a}F\left(\frac{s}{a}\right)\) |
| \(f(t-a)u(t-a)\) | \(e^{-as}F(s)\) |
| \(e^{-at}f(t)\) | \(F(s+a)\) |
| \(g'(t)\) | \(sG(s)-g(0)\) |
| \(g''(t)\) | \(s^{2}G(s)-sg(0)-g'(0)\) |
| \(g'''(t)\) | \(s^{3}G(s)-s^{2}g(0)-sg'(0)-g''(0)\) |
| \(g^{(n)}(t)\) | \(s^{n}G(s)-s^{n-1}g(0)-\cdots -g^{(n-1)}(0)\) |
| \((f\ast g)(t)=\int_{0}^{t} f(\tau )g(t-\tau )d\tau \) | \(F(s)G(s)\) |
| \(tf(t)\) | \(-F'(s)\) |
| \(t^{n}f(t)\) | \((-1)^{n}F^{(n)}(s)\) |
| \(\int_{0}^{t}f(\tau )d\tau \) | \(\frac{1}{s}F(s)\) |
| \(\frac{f(t)}{t}\) | \(\int_{s}^{\infty} F(\sigma )d\sigma\) |