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Home / Differential Equations / Laplace Transforms / 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. $$f\left( {t + T} \right) = f\left( t \right)$$ $$\displaystyle \frac{\int_{{\,0}}^{{\,T}}{{{{\bf{e}}^{ - st}}f\left( t \right)\,dt}}}}{{1 - {{\bf{e}}^{ - sT}}}$$
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

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

2. Recall the definition of hyperbolic functions. $\cosh \left( t \right) = \frac{{{{\bf{e}}^t} + {{\bf{e}}^{ - t}}}}{2}\hspace{0.25in}\hspace{0.25in}\sinh \left( t \right) = \frac{{{{\bf{e}}^t} - {{\bf{e}}^{ - t}}}}{2}$
3. 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!

4. 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

$\begin{array}{c}\Gamma \left( {p + 1} \right) = p\Gamma \left( p \right)\\ p\left( {p + 1} \right)\left( {p + 2} \right) \cdots \left( {p + n - 1} \right) =\displaystyle \frac{{\Gamma \left( {p + n} \right)}}{{\Gamma \left( p \right)}}\\ \Gamma \left( \frac{1}{2}} \right) = \sqrt \pi \end{array$
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