MATH 251 Lecture Notes - Step Function, Lu Su, Piecewise
Document Summary
The present objective is to use the laplace transform to solve differential equations with piecewise continuous forcing functions (that is, forcing functions that contain discontinuities). Before that could be done, we need to learn how to find the laplace transforms of piecewise continuous functions, and how to find their inverse transforms. Definition: the unit step function (or heaviside function), is defined by tuc. ,1 c c t t c 0. Often the unit step function uc(t) is also denoted as u(t c), hc(t), or. The step could also be negative (going down). ,0 c c t t c 0. The laplace transform of the unit step function is e cs . L{uc(t)} = s s > 0, c 0. Notice that when c = 0, u0(t) has the same laplace transform as the constant function f (t) = 1. (why?) Therefore, for our purpose, u0(t) = 1. (keep in mind that a laplace transform is only defined for t 0. )
