Differential Equations - Table Of Laplace Transforms

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Paul's Online Notes Home / Differential Equations / Laplace Transforms / Table Of Laplace Transforms Prev. Section Notes Next Section Show General Notice Show Mobile Notice Show All Notes Hide All Notes General Notice

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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

  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

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