# Step functions

### Step functions

#### Lessons

A Heaviside Step Function (also just called a “Step Function”) is a function that has a value of 0 from 0 to some constant, and then at that constant switches to 1.

The Heaviside Step Function is defined as,

The Laplace Transform of the Step Function:

$L${$u_{c}(t)$ $f(t - c)$} = $e^{-sc}$$L${$f(t)$}

$L${$u_{c}(t)$} = $\frac{e^{-sc}}{s}$

These Formulae might be necessary for shifting functions:

$\sin{(a + b)} = \sin(a)\cos(b) + \cos(a)\sin(b)$

$\cos{(a + b)} = \cos(a)\cos(b) - \sin(a)\sin(b)$

$(a + b)^{2} = a^{2} + 2ab +b^{2}$

• Introduction
a)
What is the Heaviside Step Function?

b)
What are some uses of the Heaviside Step Function and what is the Laplace Transform of a Heaviside Step Function?

• 1.
Determining Heaviside Step Functions

Write the following graph in terms of a Heaviside Step Function

• 2.
Determining the Laplace Transform of a Heaviside Step Function

Find the Laplace Transform of each of the following Step Functions:

a)
$f(t) = 6u_{3}(t) - e^{3t - 15}u_{5}(t) + 3(t - 7)^{2}u_{7}(t)$

b)
$g(t) = -\sin{(t)}u_{\pi}(t) + 2t^{2}u_{4}(t)$

• 3.
Determining the Inverse Laplace Transform of a Heaviside Step Function

Find the inverse Laplace Transform of the following function:

$F(s) = \frac{4e^{-3s}}{(s - 2)(s + 3)}$