17Calculus - Laplace Transforms
17Calculus
Laplace Transforms involve a technique to change an expression into another form that is easier to work with using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, Laplace Transforms require only improper integration techniques to use. So you may run across them in first year calculus.
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Resources
Topics You Need To Understand For This Page |
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Related Topics and Links |
Here are a couple of great playlists with many videos explaining Laplace Transforms
Michel vanBiezen - Laplace Transforms Playlist [58 videos]
video by Michel vanBiezen |
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Trefor Bazett - Laplace Transforms and Solving ODEs Playlist [12 videos]
video by Trefor Bazett |
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The Laplace Transform |
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\(\displaystyle{ \mathcal{L}\{ f(t) \} = F(s) = \int_{0}^{\infty}{e^{-st}f(t)~dt} }\) |
In the integral, \(s\) is considered a constant and \(s > 0\) |
After integration, we have a function \(F(s)\), which is a function of \(s\) (now considered a variable) |
Derivation From The Fourier Transform
Here is a great video explaining where the integral for the Laplace Transform comes from. From the video he says, 'The Laplace Transform is a generalized, one-sided weighted Fourier Transform for badly behaved functions.' So if you understand Fourier Transforms, this derivation will make sense.
Steve Brunton - The Laplace Transform: A Generalized Fourier Transform [16mins-27secs]
video by Steve Brunton |
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Overview
This takes some thinking and getting used to. So let's start off by watching some videos. There is some duplication coming up but you really need to have that repetition in order to get your head around this concept.
This first video is a quick overview with some calculations to get you started. Do not worry too much about understanding everything here since we will revisit these concepts very soon. Just get the big picture.
Dr Chris Tisdell - Intro to Laplace transform and how to calculate them [8min-48secs]
video by Dr Chris Tisdell |
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Now that you have the big picture, let's start working on understanding the details of Laplace Transforms. This is a great video clip explaining this notation and the improper integral definition. It is relatively short but is packed with good information including explaining notation and calculating the integral.
Dr Chris Tisdell - Introduction to Laplace transforms (part 1) [5min-35secs]
video by Dr Chris Tisdell |
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Okay, as explained in that video, the Laplace Transform can only be used when the improper integral converges. So the next question is, what kind of functions can the Laplace Transform be applied to?
Piecewise Continuous Functions
This video clip explains one very important type of function, a piecewise continuous function, where the Laplace Transform really helps us.
Dr Chris Tisdell - Introduction to Laplace transforms (part 2) [2min-56secs]
video by Dr Chris Tisdell |
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Plot 1 [Paul's Online Notes] |
Now, from that last video, you can tell from his graph that a piecewise continuous function is NOT a piecewise function that is continuous. It is a piecewise function that has sections that are continuous, like Plot 1. Applying the Laplace Transform to these kind of functions is where the Laplace Transform really shines.
Also notice in the last video, that he calculated the Laplace Transform of \(\cos(x)\) and \(\sin(x)\) without using the improper integral directly, which would require integration by parts. That is one of the powerful things about Laplace Transforms.
Table of Laplace Transforms
Fortunately, you are probably not going to be asked to use the improper integral at the top of the page to calculate most Laplace Transforms. You will use a Laplace Transform that you know and derive what you are looking for. In order to do that, you will usually be given a table of Laplace Transforms. These will also be used to get the inverse Laplace Transform (since no direct method exists that we can understand at this level of mathematics).
This video clip finishes the video shown previously and demonstrates how to use tables of Laplace Transforms.
Dr Chris Tisdell - Introduction to Laplace transforms (part 3) [13min-47secs]
video by Dr Chris Tisdell |
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Okay, now that you see the importance of tables for Laplace Transforms, you are probably expecting to see a table, right? Well, here is where you can find one. We highly recommend this book. It is free for you to download.
Engineering Math Workbook by Dr Chris Tisdell
Here is a list of some Laplace Transforms. We suggest you build your own list based on these, the workbook above and the pages listed in the resouces panel.
Laplace Transforms Table
Laplace Transforms | ||||
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\( f(t) \) |
\(\displaystyle{ F(s) }\) |
Practice |
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Basic Functions | ||||
\( t^n, ~ n = 1, 2, 3, \ldots \) |
\(\displaystyle{ \frac{n!}{s^{n+1}} }\) |
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\( e^{at} \) |
\(\displaystyle{ \frac{1}{s-a} }\) |
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\( \sin(\alpha t) \) |
\(\displaystyle{ \frac{\alpha}{s^2 + \alpha^2} }\) |
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\( \cos(at) \) |
\(\displaystyle{ \frac{s}{s^2 + a^2} }\) |
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\( \sinh(at) \) |
\(\displaystyle{ \frac{a}{s^2 - a^2} }\) |
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\( \cosh(at) \) |
\(\displaystyle{ \frac{s}{s^2 - a^2} }\) |
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Special Functions | ||||
\( \delta(t) \) unit impulse |
\( 1 \) |
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\( \delta(t-\tau) \) shifted unit impulse |
\( e^{-\tau s} \) |
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\( u(t) \) unit step |
\(\displaystyle{ \frac{1}{s} }\) |
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\( u(t-\tau) \) shifted unit step |
\(\displaystyle{ \frac{1}{s} e^{-\tau s} }\) |
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Combined Functions | ||||
\( e^{at}\sin(\alpha t) \) |
\(\displaystyle{ \frac{\alpha}{(s-a)^2 + \alpha^2} }\) |
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\( f(t)u(t-a) \) |
\(\displaystyle{ e^{-sa} \mathcal{L}\{ f(t+a) \} }\) |
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\( t^n e^{at}, ~ n = 1, 2, 3, \ldots \) |
\(\displaystyle{ \frac{n!}{(s-a)^{n+1}} \} }\) |
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Derivatives and Integrals | ||||
\( f'(t) \) |
\( sF(s) - f(0) \) |
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\( f''(t) \) |
\( s^2F(s) - sf(0) - f'(0) \) |
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\(\displaystyle{ f^{(n)}(t) }\) |
\(\displaystyle{ s^nF(s) - s^{n-1}f(0) - }\) \(\displaystyle{ s^{n-2}f'(0) - . . . - f^{(n-1)}(0) }\) |
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\( \int_0^t{ f(v)~dv }\) |
\(\displaystyle{ \frac{F(s)}{s} }\) |
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Examples
Okay, let's look at a couple of examples. Try them on your own first. The tools you need are
1. the definition of the Laplace Transform at the top of the page
2. improper integral techniques (Make sure you use correct notation including the limit.)
3. integration by substitution
4. integration by parts
You should know all these techniques by now, so you know how to do these calculations. The solutions are shown in video clips.
Example 1
Calculate the Laplace Transform of \(\displaystyle{ f(t) = e^{-at} }\).
This video clip shows the step-by-step solution. It also demonstrates the incredible power of the Laplace Transform. It is a little longer than the previous two video clips but make sure and watch the entire clip. Your time will be well spent.
Dr Chris Tisdell - Introduction to Laplace transforms (part 4) [21min-49secs]
video by Dr Chris Tisdell |
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Example 2
Calculate the Laplace Transform of \(tf(t)\).
Before you go to the practice problems, watch this video showing how to calculate the Laplace transform of \(tf(t)\). Yes, this probably appears your table, but it will help you get a better feel for Laplace transforms if you work through this video with the instructor. (If you feel adventurous, try working it on your own before watching the video. There is nothing new here, just an improper integral that uses integration by parts.)
Dr Chris Tisdell - Laplace Transform of tf(t) [7min-24secs]
video by Dr Chris Tisdell |
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Laplace Transform of a Periodic Function
A useful concept of the Laplace Transform is how it works with periodic functions. Periodic functions are, of course, functions that repeat in a periodic fashion. You've seen periodic functions in trigonometry. Sine and cosine are both periodic with period \(T=2\pi\).
The equation to find the Laplace Transform of a periodic function \(f(t)\) with period \(T\) is \(\displaystyle{ \mathcal{L}\{ f(t) \} = \frac{1}{1-e^{-sT}} \int_{0}^{T}{ e^{-st}f(t)~dt } }\)
Here is a great video clip showing the derivation of the last equation. Don't skip this video. It has important information that will help you when working with Laplace Transforms.
Dr Chris Tisdell - Laplace transform: square wave [8min-51secs]
video by Dr Chris Tisdell |
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Before you go on, work the practice problems. After that you need to learn how to go the other direction, i.e. when given the inverse laplace transform, how to find the original function.
Practice
Unless otherwise instructed, calculate the Laplace Transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.
\( f(t) = t \)
Problem Statement
For \( f(t) = t \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.
Solution
PatrickJMT - 650 video solution
video by PatrickJMT |
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\( f(t) = e^t \)
Problem Statement |
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Calculate the Laplace Transform of \( f(t) = e^t \) using the integral definition.
Final Answer |
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\( F(s) = \dfrac{1}{s-1} \)
Problem Statement
Calculate the Laplace Transform of \( f(t) = e^t \) using the integral definition.
Solution
PatrickJMT - 4309 video solution
video by PatrickJMT |
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Final Answer
\( F(s) = \dfrac{1}{s-1} \)
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\( y(t) = e^{at} \), \(t \geq 0 \)
Problem Statement
Calculate the Laplace Transform of \( y(t) = e^{at} \), \(t \geq 0 \) using the integral definition.
Solution
Here are two solutions from two different instructors.
Trefor Bazett - 3604 video solution
video by Trefor Bazett |
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blackpenredpen - 3604 video solution
video by blackpenredpen |
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Calculate \( \mathcal{L}\{ \sinh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).
Problem Statement
Calculate \( \mathcal{L}\{ \sinh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).
Solution
blackpenredpen - 3607 video solution
video by blackpenredpen |
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Calculate \( \mathcal{L}\{ \cosh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).
Problem Statement
Calculate \( \mathcal{L}\{ \cosh(bt) \} \) using \( \mathcal{L}\{ e^{at} \} \).
Solution
blackpenredpen - 3608 video solution
video by blackpenredpen |
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For a constant a > 0, if \( \mathcal{L}\{ f(t) \} = F(s) \), show that \(\displaystyle{ \mathcal{L}\{ f(at) \} = \frac{1}{a}F(s/a) }\)
Problem Statement |
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For a constant a > 0, if \( \mathcal{L}\{ f(t) \} = F(s) \), show that \(\displaystyle{ \mathcal{L}\{ f(at) \} = \frac{1}{a}F(s/a) }\)
Hint |
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Start by using the substitution \(u=at\) in the integral definition of the Laplace Transform.
Problem Statement
For a constant a > 0, if \( \mathcal{L}\{ f(t) \} = F(s) \), show that \(\displaystyle{ \mathcal{L}\{ f(at) \} = \frac{1}{a}F(s/a) }\)
Hint
Start by using the substitution \(u=at\) in the integral definition of the Laplace Transform.
Solution
blackpenredpen - 3615 video solution
video by blackpenredpen |
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\( \mathcal{L}\{ e^{at}f(t) \} \)
Problem Statement
\( \mathcal{L}\{ e^{at}f(t) \} \)
Solution
blackpenredpen - 3616 video solution
video by blackpenredpen |
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\( \mathcal{L}\{ e^{3t}\cos(2t) \} \)
Problem Statement
\( \mathcal{L}\{ e^{3t}\cos(2t) \} \)
Solution
blackpenredpen - 3618 video solution
video by blackpenredpen |
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\( \mathcal{L}\{ t f(t) \} \)
Problem Statement
\( \mathcal{L}\{ t f(t) \} \)
Solution
blackpenredpen - 3619 video solution
video by blackpenredpen |
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Evaluate \( \mathcal{L}\{ t \sin(bt) \} \) using a table.
Problem Statement
Evaluate \( \mathcal{L}\{ t \sin(bt) \} \) using a table.
Solution
blackpenredpen - 3620 video solution
video by blackpenredpen |
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Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using a table.
Problem Statement
Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using a table.
Solution
blackpenredpen - 3617 video solution
video by blackpenredpen |
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Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using \( \mathcal{L}\{ t^n f(t) \} = (-1)^n F^{(n)}(s) \)
Problem Statement |
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Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using \( \mathcal{L}\{ t^n f(t) \} = (-1)^n F^{(n)}(s) \)
Hint |
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\( F^{(n)}(s) \) is the nth-derivative of \( F(s) \).
Problem Statement
Evaluate \( \mathcal{L}\{ t^3 e^{2t} \} \) using \( \mathcal{L}\{ t^n f(t) \} = (-1)^n F^{(n)}(s) \)
Hint
\( F^{(n)}(s) \) is the nth-derivative of \( F(s) \).
Solution
blackpenredpen - 3621 video solution
video by blackpenredpen |
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\( f(t) = e^{3t} + \cos(6t) - e^{3t}\cos(6t) \); use a table
Problem Statement
For \( f(t) = e^{3t} + \cos(6t) - e^{3t}\cos(6t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a table.
Solution
Krista King Math - 653 video solution
video by Krista King Math |
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Find \( \mathcal{L}\{ \sin(bt) \} \) and \( \mathcal{L}\{ \cos(bt) \} \) using Euler's Formula.
Problem Statement
Find \( \mathcal{L}\{ \sin(bt) \} \) and \( \mathcal{L}\{ \cos(bt) \} \) using Euler's Formula.
Solution
blackpenredpen - 3606 video solution
video by blackpenredpen |
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\( f(t) = \sin(3t) \)
Problem Statement
For \( f(t) = \sin(3t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.
Solution
Krista King Math - 652 video solution
video by Krista King Math |
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\( f(t) = t \cosh(3t) \)
Problem Statement
For \( f(t) = t \cosh(3t) \), find the Laplace transform using the integral definition.
Solution
Krista King Math - 655 video solution
video by Krista King Math |
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\( \mathcal{L}\{ f(t)/t \} \)
Problem Statement
\( \mathcal{L}\{ f(t)/t \} \)
Solution
blackpenredpen - 3622 video solution
video by blackpenredpen |
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Evaluate \( \mathcal{L}\{ \sin(t)/t \} \) using a table.
Problem Statement
Evaluate \( \mathcal{L}\{ \sin(t)/t \} \) using a table.
Solution
blackpenredpen - 3623 video solution
video by blackpenredpen |
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\( f(t) = t^n \)
Problem Statement |
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Calculate \( \mathcal{L}\{ t^n \} \)
Hint |
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This is tricky. Two ways are to use power series or the gamma function.
Problem Statement |
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Calculate \( \mathcal{L}\{ t^n \} \)
Final Answer |
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\( \mathcal{L}\{ t^n \} = \dfrac{n!}{s^{n+1}} \)
Problem Statement
Calculate \( \mathcal{L}\{ t^n \} \)
Hint
This is tricky. Two ways are to use power series or the gamma function.
Solution
The first video uses the power series expansion of \(e^x\) and the steps to the solution are non-intuitive. The second video requires understanding of the gamma function.
blackpenredpen - 3605 video solution
video by blackpenredpen |
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Trefor Bazett - 3605 video solution
video by Trefor Bazett |
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Final Answer
\( \mathcal{L}\{ t^n \} = \dfrac{n!}{s^{n+1}} \)
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Page Sections
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Practice Instructions
Unless otherwise instructed, calculate the Laplace Transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the integral definition.
