Page 251 - Schaum's Outline of Differential Equations

234 CONVOLUTIONS AND THE UNIT STEP FUNCTION [CHAP. 23

Fig. 23-1

Theorem 23.3.

TRANSLATIONS
Given a function/(X) defined for x > 0, the function

represents a shift, or translation, of the function/(*) by c units in the positive ^-direction. For example, if f(x)
is given graphically by Fig. 23-2, then u(x- c)f(x - c) is given graphically by Fig. 23-3.

Fig. 23-2 Fig. 23-3

Theorem 23.4. If F(s) = £{f(xj}, then

Conversely,