Elementary Laplace Transforms¹

$f(t)$	$F(s)$	Notes
$f(t)$	$\rule[-16pt]{0pt}{16pt}\rule{0pt}{20pt}\displaystyle \int_{0}^{\infty} e^{- s t} f(t) \, dt$	definition
$k$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{k} \over {s}}$	$k$ is constant, $s > 0$
$e^{a t}$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{1} \over {s - a}}$	$s > a$
$t^{n}$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{n!} \over {s^{n+1}}}$	$n = 1,2,\ldots$; $s > 0$
$t^{p}$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{\Gamma(p+1)} \over {s^{p+1}}}$	$p > -1$, $s > 0$, $\Gamma$ is the gamma function
$\sin a t$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{a} \over {s^{2} + a^{2}}}$	$s > 0$
$\cos a t$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{s} \over {s^{2} + a^{2}}}$	$s > 0$
$\sinh a t$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{a} \over {s^{2} - a^{2}}}$	$s > \vert a\vert$
$\cosh a t$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{s} \over {s^{2} - a^{2}}}$	$s > \vert a\vert$
$e^{a t} \sin b t$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{b} \over {(s - a)^{2} + b^{2}}}$	$s > a$
$e^{a t} \cos b t$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{s - a} \over {(s - a)^{2} + b^{2}}}$	$s > a$
$t^{n} e^{a t}$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{n!} \over {(s - a)^{n+1}}}$	$n = 1,2,\ldots$; $s > a$
$u_{c}(t)$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{e^{- c s}} \over {s}}$	$s > 0$, $u_{c}$ is a unit step function
$u_{c}(t) f(t - c)$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle e^{- c s} F(s)$	$u_{c}$ is a unit step function
$e^{c t} f(t)$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle F(s - c)$	shifting property
$f(c t)$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle {{1} \over {c}} F \left( {{s} \over {c}} \right)$	$c > 0$
$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle \int_{0}^{t} f(t - \tau) g(\tau) \, d \tau$	$F(s) \, G(s)$	convolution integral
$\delta (t - c)$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle e^{-c s}$	$\delta$ is the Dirac delta function
$f^{(n)}(t)$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle s^{n} F(s) - s^{n-1} f(0) - \cdots - f^{(n-1)}(0)$
$(-1)^{n} f(t)$	$\rule[-14pt]{0pt}{14pt}\rule{0pt}{18pt}\displaystyle F^{(n)}(s)$

Footnotes

... Transforms¹

Adapted from

Elementary Differential Equations and Boundary Value Problems, 7th edition, William E. Boyce and Richard C. DiPrima, John Wiley & Sons, Inc., New York, 2001, p. 304