Table Of Laplace Transforms

	$f\left( t \right) = {\mathcal{L}^{\,\, - 1}}\left\{ {F\left( s \right)} \right\}$	$F\left( s \right) = \mathcal{L}\left\{ {f\left( t \right)} \right\}$
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	${f^{\left( n \right)}}\left( t \right)$	${s^n}F\left( s \right) - {s^{n - 1}}f\left( 0 \right) - {s^{n - 2}}f'\left( 0 \right) \cdots - s{f^{\left( {n - 2} \right)}}\left( 0 \right) - {f^{\left( {n - 1} \right)}}\left( 0 \right)$

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This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.