Differential Equations - Table Of Laplace Transforms

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Section 4.10 : Table Of Laplace Transforms

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.	${u_c}\left( t \right) = u\left( {t - c} \right)$ Heaviside Function	$\displaystyle \frac{{{{\bf{e}}^{ - cs}}}}{s}$
26.	$\delta \left( {t - c} \right)$ Dirac Delta Function	${{\bf{e}}^{ - cs}}$
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.	${f^{\left( n \right)}}\left( t \right)$	${s^n}F\left( s \right) - {s^{n - 1}}f\left( 0 \right) - {s^{n - 2}}f'\left( 0 \right) \cdots - s{f^{\left( {n - 2} \right)}}\left( 0 \right) - {f^{\left( {n - 1} \right)}}\left( 0 \right)$

Table Notes

This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.
Recall the definition of hyperbolic functions. $\cosh \left( t \right) = \frac{{{{\bf{e}}^t} + {{\bf{e}}^{ - t}}}}{2}\hspace{0.25in}\hspace{0.25in}\sinh \left( t \right) = \frac{{{{\bf{e}}^t} - {{\bf{e}}^{ - t}}}}{2}$
Be careful when using “normal” trig function vs. hyperbolic functions. The only difference in the formulas is the “ $+ a^{2}$ ” for the “normal” trig functions becomes a “ $- a^{2}$ ” for the hyperbolic functions!
Formula #4 uses the Gamma function which is defined as $\Gamma \left( t \right) = \int_{{\,0}}^{{\,\infty }}{{{{\bf{e}}^{ - x}}{x^{t - 1}}\,dx}}$
If $n$ is a positive integer then,
$\Gamma \left( {n + 1} \right) = n!$
The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function
$\begin{array}{c}\Gamma \left( {p + 1} \right) = p\Gamma \left( p \right)\\ p\left( {p + 1} \right)\left( {p + 2} \right) \cdots \left( {p + n - 1} \right) =\displaystyle \frac{{\Gamma \left( {p + n} \right)}}{{\Gamma \left( p \right)}}\\ \Gamma \left( {\displaystyle \frac{1}{2}} \right) = \sqrt \pi \end{array}$