Table of Laplace Transforms
- Page ID
- 20007
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| \(f(t) = \mathscr{L}^{-1}\{F(s)\}\) | \(F(s)= \mathscr{L}\{F(s)\}\) | \(f(t) = \mathscr{L}^{-1}\{F(s)\}\) | \(F(s)= \mathscr{L}\{F(s)\}\) |
|---|---|---|---|
| \(1. \quad 1\) | \(\quad \dfrac{1}{s}\) | \(2. \quad t\) | \(\quad \dfrac{1}{s^2}\) |
| \(3. \quad t^n, \; n = 1, 2, 3, \cdots\) | \(\quad \dfrac{n!}{s^{n+1}}\) | \(4. \quad e^{at}\) | \(\quad \dfrac{1}{s-a}\) |
| \(5. \quad \sin at \) | \(\quad \dfrac{a}{s^2+a^2}\) | \(6. \quad \cos at\) | \(\quad \dfrac{s}{s^2+a^2}\) |
| \(7. \quad \sinh at \) | \(\quad \dfrac{a}{s^2-a^2}\) | \(8. \quad \cosh at\) | \(\quad \dfrac{s}{s^2-a^2}\) |
| \(9. \quad e^{at}\cdot f(t)\) | \(\quad F(s - a)\) |
Unit Step or Heavyside Function |
\(\quad \dfrac{e^{-as}}{s}\) |
| \(11. \quad f(t - a)\cdot\mathscr{U}(t - a)\) | \(\quad e^{-as}\cdot F(s)\) | \(12. \quad f(t)\cdot\mathscr{U}(t - a)\) | \(\quad e^{-as}\cdot \mathscr{L}\{ f(t+a)\}\) |
| \(13. \quad f'(t) \) | \(\quad s F(s) - f(0)\) | \(14. \quad f''(t) \) | \(\quad s^2 F(s) -s\cdot f(0) - f'(0)\) |
| \(15. \quad t\cdot f(t) \) | \(\quad -F'(s)\) | \(16. \quad f^{(n)}(t) \) | \(s^n F(s) - s^{(n-1)}f(0) - \cdots\) \(- s\, f^{(n-2)}(0) - f^{(n-1)}(0)\) |
| \(17. \quad t^n\cdot f(t)\) | \(\quad (-1)^n \dfrac{d^n}{ds^n}\big(F(s)\big)\) | \(18. \quad \dfrac{1}{t}\cdot f(t) \) | \(\quad\displaystyle \int_s^\infty F(w)\,dw\) |
|
Dirac Delta Function |
\(\quad e^{-as}\) |
Convolution |
\(\quad F(s)\cdot G(s)\) |
|
Dirac Delta Function |
\(\quad f(a)\cdot e^{-as}\) |
Periodic function |
\(\displaystyle \dfrac{F_T(s)}{1-e^{-Ts}} \quad = \quad \dfrac{\int_0^T e^{-st} f(t)\, dt}{1-e^{-Ts}}\) |