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