# Formulas and Properties of Laplace Transform



## Formulas of Laplace Transform

Definition: If $f(t)$ is a one sided function such that $f(t) = 0$ for $t \lt 0$ then the Laplace transform $F(s)$ is defined by $\mathscr{L}\{f(t)\} = F(s) = \int_{0}^{+\infty} f(t) e^{-st} dt$ where $s$ is allowed to be a complex number for which the improper integral above converges.
A more precise definition of the Laplace function to accommodate for functions such as $\delta(t)$ is given by $\mathscr{L}\{f(t)\} = F(s) = \int_{0{-}}^{+\infty} f(t) e^{-st} dt$
Laplace transforms computations with examples and solutions are included.

### Transform

$f(t)$ $F(s)$
$u(t)$ $\dfrac{1}{s}$
$t^n$ $\dfrac{n!}{s^{n+1}}$
$e^{-at}$ $\dfrac{1}{s+a}$
$t^n e^{-at}$ $\dfrac{n!}{(s+a)^{n+1}}$
$\sin \omega t$ $\dfrac{\omega}{s^2+\omega^2}$
$t \sin \omega t$ $\dfrac{2 \omega s}{(s^2+\omega^2)^2}$
$\cos \omega t$ $\dfrac{s}{s^2+\omega^2}$
$t \cos \omega t$ $\dfrac{s^2 - \omega^2}{(s^2+\omega^2)^2}$
$\sinh \omega t$ $\dfrac{\omega}{s^2 - \omega^2}$
$\cosh \omega t$ $\dfrac{s }{s^2 - \omega^2}$
$\delta( t - \tau)$ $e^{-s \tau}$ , $\tau \ge 0$
$u( t - \tau)$ $\dfrac{1}{s} e^{-s \tau}$ , $\tau \ge 0$

Note
1) $\delta( t )$ is the Dirac delta function also called impulse function in engineering.
2) $u( t)$ is the Heaviside step function.

## Properties of Laplace Transform

In what follows, function $f(t)$ is written in small letters and its corresponding transform in capital letters $F(s)$
1. Linearity
If $g(t) = a f_1(t) + b f_2(t)$, then $G(s) = a F_1(s) + b F_2(s)$ , $a$ and $b$ are constants.
2. Shift in t
If $g(t) = f(t - \tau) u( t - \tau)$, then $G(s) = e^{- s \tau} F(s)$ , $\tau \ge 0$
3. Multiplication by an exponential in $t$ results in a shift in $s$
If $g(t) = e^{-at} f(t)$, then $G(s) = F(s + a)$ , $a \ge 0$
4. Scaling in $t$
If $g(t) = f(k t)$, then $G(s) = \dfrac{1}{k} F(\dfrac{s}{k})$
5. Derivative of $F(s)$ with respect to $s$
If $g(t) = t f(t)$, then $G(s) = - \dfrac{d F(s)}{d s}$
6. Derivative of $f(t)$ with respect to $t$
If $g(t) = \dfrac{df(t)}{dt} = f'(t)$, then $G(s) = s F(s) - f(0)$
7. Second derivative of $f(t)$ with respect to $t$
If $g(t) = \dfrac{df^2(t)}{dt^2} = f''(t)$, then $G(s) = s^2 F(s) - s f(0) - f'(0)$
8. $n$ th derivative of $f(t)$ with respect to $t$
If $g(t) = \dfrac{df^n(t)}{dt^n} = f^{(n)}(t)$,
then $G(s) = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - ... - s f^{(n-2)}(0) - f^{(n-1)}(0)$
9. Integral of $f(t)$ with respect to $t$
If $\displaystyle g(t) = \int_0^t f(t') dt'$ , then $G(s) = \dfrac{1}{s} F(s)$
10. Convolution integral
If $\displaystyle g(t) = \int_0^t f_1(t')f_2(t-t') dt'$, then $G(s) = F_1(s) F_2(s)$

## More References and Links

Definition of Laplace Transform .
Handbook of Mathematical Functions Page 1020.
Heaviside step function
Dirac Delta Functions
Laplace Transforms Computations Examples with Solutions.
Engineering Mathematics with Examples and Solutions