$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# Table of Laplace Transforms

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

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

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