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12: Appendix B- Table of Laplace Transforms

  • Page ID
    98111
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    The function \(u\) is the Heaviside function, \(\delta\) is the Dirac delta function, and

    \[\Gamma(t)=\int_{0}^{\infty}e^{-\tau}\tau^{t-1}d\tau ,\qquad\text{erf}(t)=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-\tau^{2}}d\tau , \qquad\text{erfc}(t)=1-\text{erf}(t).\]

    Table \(\PageIndex{1}\)

    \(f(t)\) \(F(s)=\mathcal{L}\{f(t)\}=\int_{0}^{\infty}e^{-st}f(t)dt\)
    \(C\) \(\frac{C}{s}\)
    \(t\) \(\frac{1}{s^{2}}\)
    \(t^{2}\) \(\frac{2}{s^{3}}\)
    \(t^{n}\) \(\frac{n!}{s^{n+1}}\)
    \(t^{p}\quad (p>0)\) \(\frac{\Gamma (p+1)}{s^{p+1}}\)
    \(e^{-at}\) \(\frac{1}{s+a}\)
    \(\sin(\omega t)\) \(\frac{\omega}{s^{2}+\omega ^{2}}\)
    \(\cos(\omega t)\) \(\frac{s}{s^{2}+\omega^{2}}\)
    \(\sinh (\omega t)\) \(\frac{\omega}{s^{2}-\omega ^{2}}\)
    \(\cosh (\omega t)\) \(\frac{s}{s^{2}-\omega^{2}}\)
    \(u(t-a)\) \(\frac{e^{-as}}{s}\)
    \(\delta (t)\) \(1\)
    \(\delta (t-a)\) \(e^{-as}\)
    \(\text{erf}\left(\frac{t}{2a}\right)\) \(\frac{1}{s}e^{(as)^{2}}\text{erfc}(as)\)
    \(\frac{1}{\sqrt{\pi t}}\text{exp}\left(\frac{-a^{2}}{4t}\right)\quad (a\geq 0)\) \(\frac{e^{-as}}{\sqrt{s}}\)
    \(\frac{1}{\sqrt{\pi t}}-ae^{a^{2}t}\text{erfc}(a\sqrt{t})\quad (a>0)\) \(\frac{1}{\sqrt{s}+a}\)
    \(af(t)+bg(t)\) \(aF(s)+bG(s)\)
    \(f(at)\quad (a>0)\) \(\frac{1}{a}F\left(\frac{s}{a}\right)\)
    \(f(t-a)u(t-a)\) \(e^{-as}F(s)\)
    \(e^{-at}f(t)\) \(F(s+a)\)
    \(g'(t)\) \(sG(s)-g(0)\)
    \(g''(t)\) \(s^{2}G(s)-sg(0)-g'(0)\)
    \(g'''(t)\) \(s^{3}G(s)-s^{2}g(0)-sg'(0)-g''(0)\)
    \(g^{(n)}(t)\) \(s^{n}G(s)-s^{n-1}g(0)-\cdots -g^{(n-1)}(0)\)
    \((f\ast g)(t)=\int_{0}^{t} f(\tau )g(t-\tau )d\tau \) \(F(s)G(s)\)
    \(tf(t)\) \(-F'(s)\)
    \(t^{n}f(t)\) \((-1)^{n}F^{(n)}(s)\)
    \(\int_{0}^{t}f(\tau )d\tau \) \(\frac{1}{s}F(s)\)
    \(\frac{f(t)}{t}\) \(\int_{s}^{\infty} F(\sigma )d\sigma\)

    12: Appendix B- Table of Laplace Transforms is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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