B.1 Table of Laplace Transforms

The function $u$ is the Heaviside function, $δ$ is the Dirac delta function, and

$Γ (t) = \int_{0}^{\infty} e^{- τ} τ^{t - 1} d τ, erf (t) = \frac{2}{\sqrt{π}} \int_{0}^{t} e^{- τ^{2}} d τ, erfc (t) = 1 - erf (t) .$

Table $1$

$f (t)$	$F (s) = ℒ {f (t)} = \int_{0}^{\infty} e^{- s t} f (t) d t$
$C$	$\frac{C}{s}$
$t$	$\frac{1}{s^{2}}$
$t^{2}$	$\frac{2}{s^{3}}$
$t^{n}$	$\frac{n!}{s^{n + 1}}$
$t to the power p open parenthesis p greater than 0 close parenthesis$	$\frac{Γ (p + 1)}{s^{p + 1}}$
$e^{- a t}$	$\frac{1}{s + a}$
$\sin (ω t)$	$\frac{ω}{s^{2} + ω^{2}}$
$\cos (ω t)$	$\frac{s}{s^{2} + ω^{2}}$
$\sinh (ω t)$	$\frac{ω}{s^{2} - ω^{2}}$
$\cosh (ω t)$	$\frac{s}{s^{2} - ω^{2}}$
$u (t - a)$	$\frac{e^{- a s}}{s}$
$δ (t)$	$1$
$δ (t - a)$	$e^{- a s}$
$erf (\frac{t}{2 a})$	$\frac{1}{s} e^{(a s)^{2}} erfc (a s)$
$\frac{1}{\sqrt{π t}} exp (\frac{- a^{2}}{4 t}) (a \geq 0)$	$\frac{e^{- a s}}{\sqrt{s}}$
$\frac{1}{\sqrt{π t}} - a e^{a^{2} t} erfc (a \sqrt{t}) (a > 0)$	$\frac{1}{\sqrt{s} + a}$
$a f (t) + b g (t)$	$a F (s) + b G (s)$
$f open parenthesis a t close parenthesis open parenthesis a greater than 0 close parenthesis$	$\frac{1}{a} F (\frac{s}{a})$
$f (t - a) u (t - a)$	$e^{- a s} F (s)$
$e^{- a t} f (t)$	$F (s + a)$
$g^{'} (t)$	$s G open parenthesis s close parenthesis minus g open parenthesis 0 close parenthesis$
$g^{″} (t)$	$s^{2} G (s) - s g (0) - g^{'} (0)$
$g^{‴} (t)$	$s^{3} G (s) - s^{2} g (0) - s g^{'} (0) - g^{″} (0)$
$g^{(n)} (t)$	$s^{n} G (s) - s^{n - 1} g (0) - \dots - g^{(n - 1)} (0)$
$(f * g) (t) = \int_{0}^{t} f (τ) g (t - τ) d τ$	$F (s) G (s)$
$t f (t)$	$- F^{'} (s)$
$t^{n} f (t)$	$(- 1)^{n} F^{(n)} (s)$
$∫ 0 t f open parenthesis τ close parenthesis d τ$	$\frac{1}{s} F (s)$
$\frac{f (t)}{t}$	$∫ s infinity F open parenthesis σ close parenthesis d σ$

Adapted from Differential Equations for Engineers by Jiří Lebl (Oklahoma State University), hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.