Table of Laplace Transforms

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Dec 31, 2022
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- Table of Integrals
- MTH 220 Discrete Math

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Table of Laplace Transforms
$f(t) = \mathscr{L}^{-1}\{F(s)\}$	$F(s)= \mathscr{L}\{F(s)\}$	$f(t) = \mathscr{L}^{-1}\{F(s)\}$	$F(s)= \mathscr{L}\{F(s)\}$
$1. \quad 1$	$\quad \dfrac{1}{s}$	$2. \quad t$	$\quad \dfrac{1}{s^2}$
$3. \quad t^n, \; n = 1, 2, 3, \cdots$	$\quad \dfrac{n!}{s^{n+1}}$	$4. \quad e^{at}$	$\quad \dfrac{1}{s-a}$
$5. \quad \sin at$	$\quad \dfrac{a}{s^2+a^2}$	$6. \quad \cos at$	$\quad \dfrac{s}{s^2+a^2}$
$7. \quad \sinh at$	$\quad \dfrac{a}{s^2-a^2}$	$8. \quad \cosh at$	$\quad \dfrac{s}{s^2-a^2}$
$9. \quad e^{at}\cdot f(t)$	$\quad F(s - a)$	$10. \quad \mathscr{U}(t - a)$	$\quad \dfrac{e^{-as}}{s}$
$11. \quad f(t - a)\cdot\mathscr{U}(t - a)$	$\quad e^{-as}\cdot F(s)$	$12. \quad f(t)\cdot\mathscr{U}(t - a)$	$\quad e^{-as}\cdot \mathscr{L}\{ f(t+a)\}$
$13. \quad f'(t)$	$\quad s F(s) - f(0)$	$14. \quad f''(t)$	$\quad s^2 F(s) -s\cdot f(0) - f'(0)$
$15. \quad t\cdot f(t)$	$\quad -F'(s)$	$16. \quad f^{(n)}(t)$	$s^n F(s) - s^{(n-1)}f(0) - \cdots$ $- s\, f^{(n-2)}(0) - f^{(n-1)}(0)$
$17. \quad t^n\cdot f(t)$	$\quad (-1)^n \dfrac{d^n}{ds^n}\big(F(s)\big)$	$18. \quad \dfrac{1}{t}\cdot f(t)$	$\quad\displaystyle \int_s^\infty F(w)\,dw$
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 $22. \quad f(t + T) = f(t)$	$\displaystyle \dfrac{F_T(s)}{1-e^{-Ts}} \quad = \quad \dfrac{\int_0^T e^{-st} f(t)\, dt}{1-e^{-Ts}}$

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