Table of Laplace transforms

The following is a table of Laplace transforms and inverse Laplace transforms.

Laplace transforms

$F(s) = \mathcal{L} \{f(t)\}$

$\mathcal{L} \{a f(t)\} = a \mathcal{L} \{f(t)\}$

$\mathcal{L} \{f(t) + g(t) \} = \mathcal{L} \{f(t)\} + \mathcal{L} \{g(t) \}$

$\mathcal{L} \{ f * g \} = F(s)G(s)$, f * g being the convolution integral of f and g

$\mathcal{L}\{1\}= \frac{1}{s}$

$\mathcal{L}\{t\}= \frac{1}{s^2}$

$\mathcal{L}\{t^n\} = \frac{ \Gamma (n+1)}{s^{n+1}}, \ n \ne -1$, Γ representing the gamma function

$\mathcal{L} \{ u_c (t) \} = \frac{e^{-cs}}{s}$, u_c being the Heaviside step function

$\mathcal{L} \{ u_c (t) f(t - c) \} = \frac{e^{-cs}}{s} F(s)$

$\mathcal{L}\{ \delta(t-a) \} = e^{-as}$, δ(t) being the Dirac delta function (assuming a > 0)

$\mathcal{L}\{e^{at} \}= \frac{1}{s - a}$

$\mathcal{L}\{\sin(at)\}= \frac{a}{s^2 + a^2}$

$\mathcal{L}\{\cos(at)\}= \frac{s}{s^2 + a^2}$

$\mathcal{L}\{f'(t)\}= s F(s) - f(0)$

$\mathcal{L}\{f''(t)\}= s^2 F(s) - s f(0) - f'(0)$

$\mathcal{L}\{f^n(t)\}= s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - ... - sf^{n-2} (0) - f^{n-1} (0)$

Inverse Laplace transforms

$f(t) = \mathcal{L}^{-1} \{F(s)\}$

$\mathcal{L}^{-1} \{a F(s)\} = a \mathcal{L}^{-1} \{F(s)\}$

$\mathcal{L}^{-1} \{F(s) + G(s) \} = \mathcal{L}^{-1} \{F(s)\} + \mathcal{L}^{-1} \{G(s)\}$

$\mathcal{L}^{-1} \{ F(s)G(s) \} = f * g$