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This page discusses several main functions used a lot in differential equations and easily handled using the Laplace Transform. The functions are the unit step function, unit impulse function and the square wave.

Unit Step Function

Unit Step Function

The unit step function looks exactly as the name implies. It is a discontinuous function which is piecewise continuous. It is zero to the left of zero, one to the right of zero and \(1/2\) at zero. The graph is shown on the right (\(x=t\)) and the equation is
\(\displaystyle{ u(t) = \left\{ \begin{array}{lr} 0 & t < 0 \\ 1/2 & t = 0 \\ 1 & t > 0 \end{array} \right. }\)

There are variations of this that you may see that usually involve the value at \(t=0\). Some books set this value at \(1\), others at \(0\). These variations should not affect your use of Laplace transforms. Check with your book and instructor to see which they prefer.

Looking carefully at the graph, we can see that there is only one discontinuity at \(t=0\), which is nonremovable. The graph is a function, since it passes the vertical line test. One way we could use this function is to multiply it by another function, say \(g(t)\) and, when we do that, this unit step function essentially cancels out everything to the left of zero in \(g(t)\) and everything to the right of zero stays as \(g(t)\). Using the unit step function this way is a way to filter or isolate part of a function. Now, this would be pretty limiting if everything was centered at zero. However, we can shift the unit step function to suit our needs. This is one thing shown in this first video.
Okay, let's watch a video to see how we use this function and it's Laplace transform.

Dr Chris Tisdell - Introduction to Heaviside step function [31min-5secs]

video by Dr Chris Tisdell

Unit Impulse Function

Unit Impulse Function

The unit impulse function is kind of a strange function. You can see the graph on the right. Even though we use an arrow in the graph, this is not a vector. The arrow represents the fact that the function height is infinity. We draw it from zero to one to represent that the area is one. Like the unit step function, this function can be used to filter another function. The unit impulse function filters a function at a specific point.

Formally, the unit impulse is a mathematical construct whose properties are that the area under the graph is one, with infinite height and zero width. Sounds impossible, right? Well, not in mathematics (math is cool, eh?)!

Unit Impulse Function

\(\displaystyle{ \delta (t) = \left\{ \begin{array}{lr} \infty & t = 0 \\ 0 & t \neq 0 \end{array} \right. }\)

   

\(\displaystyle{ \int_{-\infty}^{+\infty}{\delta(t)~dt} = 1 }\)

Here is a video explaining this in more detail.

Houston Math Prep - Unit Impulse & Dirac Delta Function [11min-33secs]

Square Wave

Using the formula on the main Laplace Transform page for periodic functions, this video clip shows how to use it on a square wave.

Dr Chris Tisdell - Laplace transform: square wave [2min-54secs]

video by Dr Chris Tisdell

Okay, so now you what the Laplace Transform is, how to calculate it for some simple functions and you know about some special functions that have Laplace Transforms. But what's the point? What do we use them for? You will find out very soon. However, first let's look at one other topic in Laplace Transforms, the convolution integral.

convolution integral →
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