Table of Laplace Transforms

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Table of Laplace Transforms
\(f(t) = \mathscr{L}^{-1}\{F(s)\}\)	\(F(s)= \mathscr{L}\{F(s)\}\)	\(f(t) = \mathscr{L}^{-1}\{F(s)\}\)	\(F(s)= \mathscr{L}\{F(s)\}\)
\(1. \quad 1\)	\(\quad \dfrac{1}{s}\)	\(2. \quad t\)	\(\quad \dfrac{1}{s^2}\)
\(3. \quad t^n, \; n = 1, 2, 3, \cdots\)	\(\quad \dfrac{n!}{s^{n+1}}\)	\(4. \quad e^{at}\)	\(\quad \dfrac{1}{s-a}\)
\(5. \quad \sin at \)	\(\quad \dfrac{a}{s^2+a^2}\)	\(6. \quad \cos at\)	\(\quad \dfrac{s}{s^2+a^2}\)
\(7. \quad \sinh at \)	\(\quad \dfrac{a}{s^2-a^2}\)	\(8. \quad \cosh at\)	\(\quad \dfrac{s}{s^2-a^2}\)
\(9. \quad e^{at}\cdot f(t)\)	\(\quad F(s - a)\)	Unit Step or Heavyside Function \(10. \quad \mathscr{U}(t - a)\)	\(\quad \dfrac{e^{-as}}{s}\)
\(11. \quad f(t - a)\cdot\mathscr{U}(t - a)\)	\(\quad e^{-as}\cdot F(s)\)	\(12. \quad f(t)\cdot\mathscr{U}(t - a)\)	\(\quad e^{-as}\cdot \mathscr{L}\{ f(t+a)\}\)
\(13. \quad f'(t) \)	\(\quad s F(s) - f(0)\)	\(14. \quad f''(t) \)	\(\quad s^2 F(s) -s\cdot f(0) - f'(0)\)
\(15. \quad t\cdot f(t) \)	\(\quad -F'(s)\)	\(16. \quad f^{(n)}(t) \)	\(s^n F(s) - s^{(n-1)}f(0) - \cdots\) \(- s\, f^{(n-2)}(0) - f^{(n-1)}(0)\)
\(17. \quad t^n\cdot f(t)\)	\(\quad (-1)^n \dfrac{d^n}{ds^n}\big(F(s)\big)\)	\(18. \quad \dfrac{1}{t}\cdot f(t) \)	\(\quad\displaystyle \int_s^\infty F(w)\,dw\)
Dirac Delta Function \(19. \quad \delta(t - a)\)	\(\quad e^{-as}\)	Convolution \(20. \quad f(t) * g(t)\)	\(\quad F(s)\cdot G(s)\)
Dirac Delta Function \(21. \quad \delta(t - a)f(t)\)	\(\quad f(a)\cdot e^{-as}\)	Periodic function with period T \(22. \quad f(t + T) = f(t) \)	\(\displaystyle \dfrac{F_T(s)}{1-e^{-Ts}} \quad = \quad \dfrac{\int_0^T e^{-st} f(t)\, dt}{1-e^{-Ts}}\)