Laplace Transform

The Laplace transform provides a useful method of solving certain types of differential equations when certain initial conditions are given, especially when the initial values are zero.

The Laplace transform is also very useful in the area of circuit analysis (which we see later in the Applications section) . It is often easier to analyse the circuit in its Laplace form, than to form differential equations.
The techniques of Laplace transform are not only used in circuit analysis, but also in

• Proportional-Integral-Derivative (PID) controllers
• DC motor speed control systems
• DC motor position control systems
• Second order systems of differential equations (underdamped, overdamped and critically damped)

Definition of Laplace Transform of f(t)

The Laplace transform of a function f(t) for t > 0 is defined by the following integral defined over 0 to ∞:

{ f(t)} =

The resulting expression is a function of s, which we write as F(s). In words we say

"The Laplace Transform of f(t) equals function F of s"

and write:

{f(t)} = F(s)

Similarly, the Laplace transform of a function g(t) would be written:

{g(t)} = G(s)

source: http://www.intmath.com/laplace-transformation/2-definition.php