Table of Laplace Transforms

Last updated
Save as PDF

Page ID: 33424

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

Table of Laplace Transforms
\(f(t) = \mathscr{L}^{-1}\{F(s)\}\)	\(F(s)= \mathscr{L}\{F(s)\}\)	\(f(t) = \mathscr{L}^{-1}\{F(s)\}\)	\(F(s)= \mathscr{L}\{F(s)\}\)
\(1. \quad 1\)	\(\quad \dfrac{1}{s}\)	\(2. \quad t\)	\(\quad \dfrac{1}{s^2}\)
\(3. \quad t^n, \; n = 1, 2, 3, \cdots\)	\(\quad \dfrac{n!}{s^{n+1}}\)	\(4. \quad e^{at}\)	\(\quad \dfrac{1}{s-a}\)
\(5. \quad \sin at \)	\(\quad \dfrac{a}{s^2+a^2}\)	\(6. \quad \cos at\)	\(\quad \dfrac{s}{s^2+a^2}\)
\(7. \quad \sinh at \)	\(\quad \dfrac{a}{s^2-a^2}\)	\(8. \quad \cosh at\)	\(\quad \dfrac{s}{s^2-a^2}\)
\(9. \quad e^{at}\cdot f(t)\)	\(\quad F(s - a)\)	Unit Step or Heavyside Function \(10. \quad \mathscr{U}(t - a)\)	\(\quad \dfrac{e^{-as}}{s}\)
\(11. \quad f(t - a)\cdot\mathscr{U}(t - a)\)	\(\quad e^{-as}\cdot F(s)\)	\(12. \quad f(t)\cdot\mathscr{U}(t - a)\)	\(\quad e^{-as}\cdot \mathscr{L}\{ f(t+a)\}\)
\(13. \quad f'(t) \)	\(\quad s F(s) - f(0)\)	\(14. \quad f''(t) \)	\(\quad s^2 F(s) -s\cdot f(0) - f'(0)\)
\(15. \quad t\cdot f(t) \)	\(\quad -F'(s)\)	\(16. \quad f^{(n)}(t) \)	\(s^n F(s) - s^{(n-1)}f(0) - \cdots\) \(- s\, f^{(n-2)}(0) - f^{(n-1)}(0)\)
\(17. \quad t^n\cdot f(t)\)	\(\quad (-1)^n \dfrac{d^n}{ds^n}\big(F(s)\big)\)	\(18. \quad \dfrac{1}{t}\cdot f(t) \)	\(\quad\displaystyle \int_s^\infty F(w)\,dw\)
Dirac Delta Function \(19. \quad \delta(t - a)\)	\(\quad e^{-as}\)	Convolution \(20. \quad f(t) * g(t)\)	\(\quad F(s)\cdot G(s)\)
Dirac Delta Function \(21. \quad \delta(t - a)f(t)\)	\(\quad f(a)\cdot e^{-as}\)	Periodic function with period T \(22. \quad f(t + T) = f(t) \)	\(\displaystyle \dfrac{F_T(s)}{1-e^{-Ts}} \quad = \quad \dfrac{\int_0^T e^{-st} f(t)\, dt}{1-e^{-Ts}}\)

Search

Text Color

Text Size

Margin Size

Font Type