The Laplace Trans! arm Method 259Jump discontinuities occur in physical problems such as electrical circuitswhich include "on/off" switches. Consider the function g(x) = u(x -c)f(x).
When x < c, u(x-c) = 0, and therefore g(x) = 0. When x 2: c, u(x-c) = 1,
and consequently g(x) = f(x). That is,
{0,g(x) = u(x - c)f(x) =
j(x), C~Xx<c
Hence, multiplication of the function f(x) by the unit step function u(x - c)"turns off" the function for x < c and "turns on" t he function for x 2: c.
Graphs of y(x) =cos x and y(x) = u(x - n) cos x are displayed in Figure 5.4.
y(x) =cos x
~
-2 ./""- x
.-1.5y(x) = u(x -n:) cos x
2 'T
~x
2Figure5.4 Graphsof y(x)= cosx and y(x)=u(x-n)cosx
Observe that the piecewise function{f(x),h(x) =
g(x),can be written using unit step functions as
x<c
c~xh(x) = f(x) -u(x - c)f(x) + u(x - c)g(x).
For a ~ x < b the "filter" function [u(x-a)-u(x-b)] when multiplying the
function f(x) "turns off" the function f(x) for x <a, "turns on" the function
f(x) for a~ x < b, and "turns off" the function f(x) for b ~ x. That is,
l
0, x <a
[u(x - a) -u(x -b)]f(x) = f(x), a~ x < b
0, b ~ xGraphs of cosx, [u(x -n) -u(x - 2n)], and [u(x - n) - u(x - 2n)] cosx are
shown in Figure 5. 5.