# 8.4: A Brief Table of Laplace Transforms

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

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

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

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

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

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

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

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

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

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

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

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

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

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

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

Table $$\PageIndex{1}$$
$$\displaystyle f(t)$$ $$\displaystyle F(s)$$
1 $$\displaystyle{ 1\over s}$$ $$\displaystyle (s > 0)$$
$$\displaystyle t^n$$ $$\displaystyle{ n!\over s^{n+1} }$$ $$\displaystyle (s > 0)$$
($$\displaystyle n = \mbox{ integer } > 0$$)
$$\displaystyle t^p,\; p > -1$$ $$\displaystyle{ \Gamma (p+1) \over s^{(p+1)} }$$ $$\displaystyle (s>0)$$
$$\displaystyle e^{at}$$ $$\displaystyle{ 1 \over s-a }$$ $$\displaystyle (s > a)$$
$$\displaystyle t^ne^{at}$$ $$\displaystyle{ n! \over (s-a)^{n+1} }$$ $$\displaystyle (s > 0)$$
($$\displaystyle n= \text{ integer } > 0$$)
$$\displaystyle \cos \omega t$$ $$\displaystyle{ \frac{s}{s^{2}+\omega ^{2}} }$$ $$\displaystyle (s > 0)$$
$$\displaystyle \sin \omega t$$ $${ \displaystyle \omega \over s^2+\omega^2 }$$ $$\displaystyle (s > 0)$$
$$\displaystyle e^{\lambda t} \cos \omega t$$ $$\displaystyle{ s - \lambda \over (s-\lambda)^2+\omega^2 }$$ $$\displaystyle (s > \lambda)$$
$$\displaystyle e^{\lambda t} \sin \omega t$$ $$\displaystyle{ \omega \over (s-\lambda)^2+\omega^2 }$$ $$\displaystyle (s > \lambda)$$
$$\displaystyle \cosh bt$$ $$\displaystyle{ s \over s^2-b^2 }$$ $$\displaystyle (s > |b|)$$
$$\displaystyle \sinh bt$$ $$\displaystyle{ b \over s^2-b^2 }$$ $$\displaystyle (s > |b|)$$
$$\displaystyle t \cos \omega t$$ $$\displaystyle{ s^2-\omega^2 \over (s^2+\omega^2)^2 }$$ $$\displaystyle (s>0)$$
$$\displaystyle t \sin \omega t$$ $$\displaystyle{ 2\omega s \over (s^2+\omega^2)^2 }$$ $$\displaystyle (s>0)$$
$$\displaystyle \sin \omega t -\omega t\cos \omega t$$ $$\displaystyle{ 2\omega^3\over (s^2+\omega^2)^2 }$$ $$\displaystyle (s>0)$$
$$\displaystyle \omega t - \sin \omega t$$ $$\displaystyle{ \omega^3 \over s^2(s^2+\omega^2) }$$ $$\displaystyle (s>0)$$
$$\displaystyle \frac{1}{t}\sin\omega t$$ $$\displaystyle{ \arctan \left({\omega \over s}\right) }$$ $$\displaystyle (s>0)$$
$$\displaystyle e^{at}f(t)$$ $$\displaystyle{ F(s-a) }$$
$$\displaystyle t^kf(t)$$ $$\displaystyle (-1)^{k}F^{(k)}(s)$$
$$\displaystyle f(\omega t)$$ $$\displaystyle{ \frac{1}{\omega}F\left(\frac{s}{\omega } \right), \quad \omega >0 }$$
$$\displaystyle u(t-\tau)$$ $$\displaystyle{ e^{-\tau s} \over s }$$ $$\displaystyle (s>0)$$
$$\displaystyle u(t-\tau)f(t-\tau)\, (\tau > 0)$$ $$\displaystyle{ e^{-\tau s}F(s) }$$
$$\displaystyle \displaystyle {\int^t_o f(\tau)g(t-\tau)\, d\tau}$$ $$\displaystyle{ F(s) \cdot G(s) }$$
$$\displaystyle \delta(t-a)$$ $$\displaystyle{ e^{-as} }$$ $$\displaystyle (s>0)$$

This page titled 8.4: A Brief Table of Laplace Transforms is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.