Laplace transform

The Laplace Transform of a function is a linear transformation from the space of $\Re \to \Re$ ( $\Re$ is the space of integratable functions) defined as

$\pounds \{ f \} (s) = F(s) = \int _{0} ^ {\infty} e^{-st} f(t) dt$

Uses

The Laplace Transform is a technique used to solve differential equation when some of the coefficients are not continuous functions.

We first take the Laplace Transform of the equation, then solve the resulting algebra equation for $Y(s)$ , and then take the inverse Laplace Transform of $Y(s)$ to get $y$ .

Here is a convenient table for reference:

Table of Laplace Tranforms
1. $1$	$\frac{1}{s}, s>0$
2. $e^{at}$	$\frac{1}{s-a}, s>a$
3. $t^n, \text{n is a positive integer}$	$\frac{n!}{s^{n+1}},s>0$
4. $t^p, p>-1$	$\frac{\Gamma (p+1)}{s^{p+1}}, s>0$
5. $\sin (at)$	$\frac{a}{s^2+a^2},s>0$
6. $\cos (at)$	$\frac{s}{s^2+a^2}, s>0$
7. $\sinh (at)$	$\frac{a}{s^2-a^2}, s>\|a\|$
8. $\cosh (at)$	$\frac{s}{s^2-a^2}, s>\|a\|$
9. $\text{Heaviside} (t-a) = \begin{cases} 1 &\text{if } t \ge a \\ 0 &\text{if } t < a \end{cases}$	$\frac{e^{-as}}{s}, s>0$
10. $\text{Heaviside} (t-a) \cdot f(t-a)$	$F(s-a)$