transforms

Laplace Transform

exponential function

by thinking of the derivative of F (x) as a ratio of increments. The increment at

the numerator is given by the difference of two areas obtained by shifting the

right edge by dx. The increment at the denominator is dx itself. Called m the

average value taken by f () in the interval having length dx, such value converges

to f (x) as dx approaches zero.

fact that the derivative of a constant is zero, and it justifies the fact that the po-

sition of the first integration edge doesn't come into play in the relationship (44)

between a function and its primitive.

two edges a and b by the formula

of derivation and integration, the reader without a background in calculus is

referred to chapter VIII of the book [25].

tions. Mathematicians have always tried to find alternative ways of expressing

functions and operations on them. This research has expressed some transforms

which, in many cases, allow to study and manipulate some classes of functions

more easily.

The Laplace transform of a function y(t), t R is defined as a function of the

complex variable s:

a vertical strip in the complex plane, and within this strip the transform can be

inverted with

along a vertical line with abscissa .