# 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 $$G(s) = F_1(s) F_2(s)$$