Table of Laplace Transforms
- Page ID
- 20007
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\(f(t) = \mathscr{L}^{-1}\{F(s)\}\) | \(F(s)= \mathscr{L}\{F(s)\}\) | \(f(t) = \mathscr{L}^{-1}\{F(s)\}\) | \(F(s)= \mathscr{L}\{F(s)\}\) |
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\(1. \quad 1\) | \(\quad \dfrac{1}{s}\) | \(2. \quad t\) | \(\quad \dfrac{1}{s^2}\) |
\(3. \quad t^n, \; n = 1, 2, 3, \cdots\) | \(\quad \dfrac{n!}{s^{n+1}}\) | \(4. \quad e^{at}\) | \(\quad \dfrac{1}{s-a}\) |
\(5. \quad \sin at \) | \(\quad \dfrac{a}{s^2+a^2}\) | \(6. \quad \cos at\) | \(\quad \dfrac{s}{s^2+a^2}\) |
\(7. \quad \sinh at \) | \(\quad \dfrac{a}{s^2-a^2}\) | \(8. \quad \cosh at\) | \(\quad \dfrac{s}{s^2-a^2}\) |
\(9. \quad e^{at}\cdot f(t)\) | \(\quad F(s - a)\) |
Unit Step or Heavyside Function |
\(\quad \dfrac{e^{-as}}{s}\) |
\(11. \quad f(t - a)\cdot\mathscr{U}(t - a)\) | \(\quad e^{-as}\cdot F(s)\) | \(12. \quad f(t)\cdot\mathscr{U}(t - a)\) | \(\quad e^{-as}\cdot \mathscr{L}\{ f(t+a)\}\) |
\(13. \quad f'(t) \) | \(\quad s F(s) - f(0)\) | \(14. \quad f''(t) \) | \(\quad s^2 F(s) -s\cdot f(0) - f'(0)\) |
\(15. \quad t\cdot f(t) \) | \(\quad -F'(s)\) | \(16. \quad f^{(n)}(t) \) | \(s^n F(s) - s^{(n-1)}f(0) - \cdots\) \(- s\, f^{(n-2)}(0) - f^{(n-1)}(0)\) |
\(17. \quad t^n\cdot f(t)\) | \(\quad (-1)^n \dfrac{d^n}{ds^n}\big(F(s)\big)\) | \(18. \quad \dfrac{1}{t}\cdot f(t) \) | \(\quad\displaystyle \int_s^\infty F(w)\,dw\) |
Dirac Delta Function |
\(\quad e^{-as}\) |
Convolution |
\(\quad F(s)\cdot G(s)\) |
Dirac Delta Function |
\(\quad f(a)\cdot e^{-as}\) |
Periodic function |
\(\displaystyle \dfrac{F_T(s)}{1-e^{-Ts}} \quad = \quad \dfrac{\int_0^T e^{-st} f(t)\, dt}{1-e^{-Ts}}\) |