2.1 The Laplace Transform

The Laplace transform represents a powerful tool for solving a wide variety of problems (differential equations, etc.). The Laplace transform permits us to replace the derivative operator d/ dt in differential equations by s, which leads to the use of algebraic methods. This is as if we had to carry out multiplication of roman numerals, and we converted them first to arabic numerals. A function f ( t) of a variable t is called original if the following conditions are fulfilled:

1. f( t) = 0 for t < 0;

2. f ( t) is piecewise ^[1] continuous on [0, ?[;

3. f ( t) is of exponential order (it has bounded exponential growth), i.e., f(t) ? M exp (s ₀t) or is called the indicator.

The one-sided Laplace transform ^[2] (image) of the function f ( t) is denoted by F ( s) or and is given by:

which corresponds to an integral transformation. This integral is convergent. On the basis of the third condition above, we obtain:

In other words, the integral (2.1) does not converge for functions that increase faster than exponentially. We can easily verify that the Laplace transform of f ( t) = exp ( t ²) does not exist. Indeed, exp ( t ²) > M exp ( t). From the...