• <blockquote id="a6oys"></blockquote>
  • <tt id="a6oys"></tt>
  • Paul's Online Notes
    Paul's Online Notes
    Home / Differential Equations / Laplace Transforms / Table Of Laplace Transforms
    Show General Notice Show Mobile Notice Show All Notes Hide All Notes
    General Notice

    I apologize for the outage on the site yesterday and today. Lamar University is in Beaumont Texas and Hurricane Laura came through here and caused a brief power outage at Lamar. Things should be up and running at this point and (hopefully) will stay that way, at least until the next hurricane comes through here which seems to happen about once every 10-15 years. Note that I wouldn't be too suprised if there are brief outages over the next couple of days as they work to get everything back up and running properly. I apologize for the inconvienence.

    Paul
    August 27, 2020

    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

      \[\begin{array}{c}\Gamma \left( {p + 1} \right) = p\Gamma \left( p \right)\\ p\left( {p + 1} \right)\left( {p + 2} \right) \cdots \left( {p + n - 1} \right) =\displaystyle \frac{{\Gamma \left( {p + n} \right)}}{{\Gamma \left( p \right)}}\\ \Gamma \left( {\displaystyle \frac{1}{2}} \right) = \sqrt \pi \end{array}\]
    亚洲欧美中文日韩视频 - 视频 - 在线观看 - 影视资讯 - av网 <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <蜘蛛词>| <文本链> <文本链> <文本链> <文本链> <文本链> <文本链>