The Laplace transform is a well established mathematical technique for solving differential equations. It is named in honor of the great French mathematician, Pierre Simon De Laplace (1749-1827). Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations.

Many mathematical problems are solved using transformations. The idea is to transform the problem into another problem that is easier to solve. Once a solution is obtained, the inverse transform is used to obtain the solution to the original problem. The Laplace transform is an important tool that makes a solution of linear constant coefficient differential equations much easier.

The Laplace transform transforms the differential equations into algebraic equations which are easier to manipulate and solve. Once the solution is obtained in the Laplace transform domain is obtained, the inverse transform is used to obtain the solution to the differential equation. Laplace transform is an essential tool for the study of linear time-invariant systems.

The Laplace transform allows the time domain to be complex; however, this is seldom needed in signal processing. In this discussion and nearly all practical applications, the time domain signal is completely real.

Definition

Let f(t) be given for t ≥ 0 and assume the function satisfy certain conditions to be stated later on.

The Laplace transform of f(t), that it is denoted by

\({f(t)}\;or\;F(s)\)

is defined by the equation

\(F\left ( s \right )=\int_{o }^{\infty} f\left ( t \right ) e^{-st} dt\)

whenever the improper integral converges. Remark: The kernel of the Laplace transform is \(K \left ( s,t \right )= e^{-st}\)<

Let’s compute the of Laplace transform of some important elementary functions, before discussing the restriction that have to be imposed on f(t) so it has a Laplace transform.

**Table of Laplace Transforms**