Step Functions; and Laplace Transforms of Piecewise ...

Step Functions; and Laplace Transforms of Piecewise Continuous Functions

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. Our starting point is to study how a piecewise continuous function can be constructed using step functions. Then we will see how the Laplace transform and its inverse interact with the said construct.

Step Functions

Definition: The unit step function (or Heaviside function), is defined by

uc

( t

)

=

0, 1,

t 0.

? 2008 Zachary S Tseng

C-2 - 2

The unit step function is much more useful than it first appears to be. When put in a product with a second function, the unit step function acts like a switch to turn the other function on or off:

uc

(t

)

f

(t

)

=

0, f (t

),

t ................
................

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