Page 250 - Schaum's Outline of Differential Equations

CHAPTER 23

Convolutions and

the Unit Step Function

CONVOLUTIONS

The convolution of t\vo functions/^) and x(x) is

Theorem 23.1. f i x ) * g (x) = g ix) */(.v).

Theorem 23.2. (Convolution theorem).

It follows directly from these two theorems that

If one of the t"o comolmions in Hq. (23. 2) is simpler 10 calculate, then thai convolution is chosen \vhen
determining the inverse Laplace transform of a product.

UNIT STEP FUNCTION
The unit step function u(x) is defined as

<
As an immediate consequence of the definition, we have for any number .

The graph of u(x — r) is given if Fig. 23- 1 .

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