Step Functions

In this discussion, we will investigate piecewise defined functions and their Laplace Transforms.  We start with the fundamental piecewise defined function, the Heaviside function.

 Definition:  The Heaviside function The Heaviside function, also called the unit step function, is defined by

The Heaviside function y  =  uc(t) and y = 1 - uc(t) are graphed below.

Example

We can write the function

In terms of Heaviside functions.

Solution

We tackle the functions in parts.  The function that is 1 from 0 to 2 and 0 otherwise is

1 - u2(x)

Multiplying by 3 gives

3(1 - u2(x))  =  3 - 3u2(x)

To get the function that is 1 between 2 and 5 and 0 otherwise, we subtract

u2(x) - u5(x)

Now multiply by ex to get

ex(u2(x) - u5(x))  =  ex u2(x) - ex u5(x)

f(x)  =  3 - 3u2(x) + ex u2(x) - ex u5(x)

=  3 + (ex - 3)u2(x) - ex u5(x)

We can find the Laplace transform of uc(t) by integrating

 e-cs           L{uc(t)}  =                                                             s

In practice, we want to find the Laplace transform of a more general piecewise defined function such as

This type of function occurs in electronics when a switch is suddenly turned on after one second and a forcing function is applied.  We can write

f(x)  =  up(x) sin x

We will be interested in the Laplace transform of a product of the Heaviside function with a continuous function.  The result that we need is

 L{uc(t) f(t - c)}  =  e-cs L{f(t)}

By taking inverses we get that if F(s)  =  L{f(t)}, then

L-1{e-csF(s)} =  uc(t)f(t - c)

Proof

We use the definition to get

Example

Find the Laplace transform of

Solution

We use that fact that

f(x)  =  up(x)sin x  =  -up(x) sin[(x - p)]

Now we can use the formula to get that

L{f(x)}  =  -L{up(x) sin[(x - p)]}  =  -e-cs L{sin x}

By the table, we get

-e-cs

=
s2 + 1