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Mathematics LibreTexts

Table of Laplace Transforms

  • Page ID
    20007
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    Table of Laplace Transforms
    \(f(t) = \mathscr{L}^{-1}\{F(s)\}\) \(F(s)= \mathscr{L}\{F(s)\}\) \(f(t) = \mathscr{L}^{-1}\{F(s)\}\) \(F(s)= \mathscr{L}\{F(s)\}\)
    \(1. \quad 1\) \(\quad \dfrac{1}{s}\) \(2. \quad t\) \(\quad \dfrac{1}{s^2}\)
    \(3. \quad t^n, \; n = 1, 2, 3, \cdots\) \(\quad \dfrac{n!}{s^{n+1}}\) \(4. \quad e^{at}\) \(\quad \dfrac{1}{s-a}\)
    \(5. \quad \sin at \) \(\quad \dfrac{a}{s^2+a^2}\) \(6. \quad \cos at\) \(\quad \dfrac{s}{s^2+a^2}\)
    \(7. \quad \sinh at \) \(\quad \dfrac{a}{s^2-a^2}\) \(8. \quad \cosh at\) \(\quad \dfrac{s}{s^2-a^2}\)
    \(9. \quad e^{at}\cdot f(t)\) \(\quad F(s - a)\)

    \(10. \quad \mathscr{U}(t - a)\)

    Unit Step or Heavyside Function

    \(\quad \dfrac{e^{-as}}{s}\)
    \(11. \quad f(t - a)\cdot\mathscr{U}(t - a)\) \(\quad e^{-as}\cdot F(s)\) \(12. \quad f(t)\cdot\mathscr{U}(t - a)\) \(\quad e^{-as}\cdot \mathscr{L}\{ f(t+a)\}\)
    \(13. \quad f'(t) \) \(\quad s F(s) - f(0)\) \(14. \quad f''(t) \) \(\quad s^2 F(s) -s\cdot f(0) - f'(0)\)
    \(15. \quad t\cdot f(t) \) \(\quad -F'(s)\) \(16. \quad f^{(n)}(t) \) \(s^n F(s) - s^{(n-1)}f(0) - \cdots\)
    \(- s\, f^{(n-2)}(0) - f^{(n-1)}(0)\)
    \(17. \quad t^n\cdot f(t)\) \(\quad (-1)^n \dfrac{d^n}{ds^n}\big(F(s)\big)\) \(18. \quad \dfrac{1}{t}\cdot f(t) \) \(\quad\displaystyle \int_s^\infty F(w)\,dw\)

    \(19. \quad \delta(t - a)\)

    Dirac Delta Function

    \(\quad e^{-as}\)

    \(20. \quad f(t) * g(t)\)

    Convolution

    \(\quad F(s)\cdot G(s)\)

    \(21. \quad f(t + T) = f(t) \)

    Periodic function
    with period T

    \(\displaystyle \dfrac{F_T(s)}{1-e^{-Ts}} \quad = \quad \dfrac{\int_0^T e^{-st} f(t)\, dt}{1-e^{-Ts}}\)