| $f\left( t \right) = {\mathcal{L}^{\,\, - 1}}\left\{ {F\left( s \right)} \right\}$ | $F\left( s \right) = \mathcal{L}\left\{ {f\left( t \right)} \right\}$ | |
|---|---|---|
| 1 | $1$ | $\displaystyle \frac{1}{s}$ |
| 2 | ${{\bf{e}}^{a\,t}}$ | $\displaystyle \frac{1}{{s - a}}$ |
| 3 | ${t^n},\,\,\,\,\,n = 1,2,3, \ldots$ | $\displaystyle \frac{{n!}}{{{s^{n + 1}}}}$ |
| 4 | ${t^p},\,\,\,\,\,p > -1$ | $\displaystyle \frac{{\Gamma \left( {p + 1} \right)}}{{{s^{p + 1}}}}$ |
| 5 | $\sqrt t$ | $\displaystyle \frac{{\sqrt \pi }}{{2{s^{\frac{3}{2}}}}}$ |
| 6 | ${t^{n - \frac{1}{2}}},\,\,\,\,\,n = 1,2,3, \ldots$ | $\displaystyle \frac{{1 \cdot 3 \cdot 5 \cdots \left( {2n - 1} \right)\sqrt \pi }}{{{2^n}{s^{n + \frac{1}{2}}}}}$ |
| 7 | $\sin \left( {at} \right)$ | $\displaystyle \frac{a}{{{s^2} + {a^2}}}$ |
| 8 | $\cos \left( {at} \right)$ | $\displaystyle \frac{s}{{{s^2} + {a^2}}}$ |
| 9 | $t\sin \left( {at} \right)$ | $\displaystyle \frac{{2as}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$ |
| 10 | $t\cos \left( {at} \right)$ | $\displaystyle \frac{{{s^2} - {a^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$ |
| 11 | $\sin \left( {at} \right) - at\cos \left( {at} \right)$ | $\displaystyle \frac{{2{a^3}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$ |
| 12 | $\sin \left( {at} \right) + at\cos \left( {at} \right)$ | $\displaystyle \frac{{2a{s^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$ |
| 13 | $\cos \left( {at} \right) - at\sin \left( {at} \right)$ | $\displaystyle \frac{{s\left( {{s^2} - {a^2}} \right)}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$ |
| 14 | $\cos \left( {at} \right) + at\sin \left( {at} \right)$ | $\displaystyle \frac{{s\left( {{s^2} + 3{a^2}} \right)}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}$ |
| 15 | $\sin \left( {at + b} \right)$ | $\displaystyle \frac{{s\sin \left( b \right) + a\cos \left( b \right)}}{{{s^2} + {a^2}}}$ |
| 16 | $\cos \left( {at + b} \right)$ | $\displaystyle \frac{{s\cos \left( b \right) - a\sin \left( b \right)}}{{{s^2} + {a^2}}}$ |
| 17 | $\sinh \left( {at} \right)$ | $\displaystyle \frac{a}{{{s^2} - {a^2}}}$ |
| 18 | $\cosh \left( {at} \right)$ | $\displaystyle \frac{s}{{{s^2} - {a^2}}}$ |
| 19 | ${{\bf{e}}^{at}}\sin \left( {bt} \right)$ | $\displaystyle \frac{b}{{{{\left( {s - a} \right)}^2} + {b^2}}}$ |
| 20 | ${{\bf{e}}^{at}}\cos \left( {bt} \right)$ | $\displaystyle \frac{{s - a}}{{{{\left( {s - a} \right)}^2} + {b^2}}}$ |
| 21 | ${{\bf{e}}^{at}}\sinh \left( {bt} \right)$ | $\displaystyle \frac{b}{{{{\left( {s - a} \right)}^2} - {b^2}}}$ |
| 22 | ${{\bf{e}}^{at}}\cosh \left( {bt} \right)$ | $\displaystyle \frac{{s - a}}{{{{\left( {s - a} \right)}^2} - {b^2}}}$ |
| 23 | ${t^n}{{\bf{e}}^{at}},\,\,\,\,\,n = 1,2,3, \ldots$ | $\displaystyle \frac{{n!}}{{{{\left( {s - a} \right)}^{n + 1}}}}$ |
| 24 | $f\left( {ct} \right)$ | $\displaystyle \frac{1}{c}F\left( {\frac{s}{c}} \right)$ |
| 25 | ${u_c}\left( t \right) = u\left( {t - c} \right)$ | $\displaystyle \frac{{{{\bf{e}}^{ - cs}}}}{s}$ |
| 26 | $\delta \left( {t - c} \right)$ | ${{\bf{e}}^{ - cs}}$ |
| 27 | ${u_c}\left( t \right)f\left( {t - c} \right)$ | ${{\bf{e}}^{ - cs}}F\left( s \right)$ |
| 28 | ${u_c}\left( t \right)g\left( t \right)$ | ${{\bf{e}}^{ - cs}}{\mathcal{L}}\left\{ {g\left( {t + c} \right)} \right\}$ |
| 29 | ${{\bf{e}}^{ct}}f\left( t \right)$ | $F\left( {s - c} \right)$ |
| 30 | ${t^n}f\left( t \right),\,\,\,\,\,n = 1,2,3, \ldots$ | ${\left( { - 1} \right)^n}{F^{\left( n \right)}}\left( s \right)$ |
| 31 | $\displaystyle \frac{1}{t}f\left( t \right)$ | $\int_{{\,s}}^{{\,\infty }}{{F\left( u \right)\,du}}$ |
| 32 | $\displaystyle \int_{{\,0}}^{{\,t}}{{\,f\left( v \right)\,dv}}$ | $\displaystyle \frac{{F\left( s \right)}}{s}$ |
| 33 | $\displaystyle \int_{{\,0}}^{{\,t}}{{f\left( {t - \tau } \right)g\left( \tau \right)\,d\tau }}$ | $F\left( s \right)G\left( s \right)$ |
| 34 | $f\left( {t + T} \right) = f\left( t \right)$ | $\displaystyle \frac{{\displaystyle \int_{{\,0}}^{{\,T}}{{{{\bf{e}}^{ - st}}f\left( t \right)\,dt}}}}{{1 - {{\bf{e}}^{ - sT}}}}$ |
| 35 | $f'\left( t \right)$ | $sF\left( s \right) - f\left( 0 \right)$ |
| 36 | $f''\left( t \right)$ | ${s^2}F\left( s \right) - sf\left( 0 \right) - f'\left( 0 \right)$ |
| 37 | ${f^{\left( n \right)}}\left( t \right)$ | ${s^n}F\left( s \right) - {s^{n - 1}}f\left( 0 \right) - {s^{n - 2}}f'\left( 0 \right) \cdots - s{f^{\left( {n - 2} \right)}}\left( 0 \right) - {f^{\left( {n - 1} \right)}}\left( 0 \right)$ |
This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.