Pauls Online Notes
Pauls Online Notes
Home / Differential Equations / Laplace Transforms / Table Of Laplace Transforms
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

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

  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

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