The Laplace Transform: A Generalized Fourier Transform

Welcome back, I'm Steve Brenton and today I'm going to talk to you about the Laplace

transform

, which is one of my favorite

transform

ations in all of mathematics. The Laplace transform. Many of you have heard of the Fourier transform. I just gave a full lecture. series on the Fourier transform, the fast Fourier transform, and I think the Laplace transform is the culmination of all the work on the Fourier transform, so Laplace, in a sense,

generalized

the Fourier transform to a class of functions much bigger and more important than you now I can transform, ok, so I'll tell you all about that right now.

I'll explain how to derive the Fourier transform from the Fourier transform and then how to use it to do things like calculate derivatives and solve equations. I will point out that you know that in mathematics there is generally no magic wand, but the Laplace transform is as close as it gets: you can take a system and subtract about 2 or 3 years of advanced mathematics from how difficult that system is to solve. by simply applying the Laplace transform, for example, if you have a partial differential equation, a PDE, under certain circumstances, you can Laplace transform it and convert it from a PDE to an OD e, which is much simpler, similarly, you can take an OD e and under some conditions, you can transform it with the Laplace transform into an algebraic equation, which again ranges from college to high school as solution techniques and the Laplace transform is also extremely useful in control theory, for so the Laplace transform will emerge everywhere. and today I'm going to derive it for you and show you how it's actually not separate from the Fourier transform; in a sense, it is a

generalized

Fourier transform.

Well, many of you have seen the Laplace transform and the Fourier transform for years and today I have noticed similarities. I'm going to show you how they are exactly the same. The Laplace transform comes from the Fourier transform, so the plus is one of my favorite French mathematicians. He was the son of a farmer and he got his name on the Eiffel Tower, one of the things I love about the French is that they venerate their great thinkers, mathematicians and scientists, and the square was really one of the funniest facts about Laplace. He was one of the first researchers to realize this. that when you're dealing with real world data that's noisy and not perfect, you have to look at that data through a probabilistic lens through the lens of probability theory, for us we take that for granted, but that was a big problem in the past. when plos lived in the latter half of the 1700s and early 1800s, okay, so let's jump to the big idea here: we know that we can Fourier transform into nice, well-behaved functions that decay to zero at plus and minus infinity, like this I'm just going to give an example of that, if I have this nice Gaussian that goes to zero when X or T goes to plus or minus infinity, I can do Fourier transformations.

I will say that the Fourier transform verifies that we can do the Fourier transform, but less well-behaved functions, so there are nastier functions and I'm going to draw a couple of them right now, like e to the lambda T. This function is not can Fourier transform because it doesn't go to 0 when T goes to 2. plus infinity, so you can't Fourier transform this function. Another example of a function that is complicated or impossible to Fourier transform is the Heaviside function. I really like this Heaviside function which is 0 for negative time and 1 for positive time, so let's be explicit and call this. time and this is the Heaviside function named after oliver heaviside and is 0 for T less than 0 and is 1 for T greater than or equal to 0.

This function also cannot be easily Fourier transformed because it does not gradually decrease to 0 in plus infinity ok now there are other examples where you can technically Fourier transform it's a bit annoying so let's think about this trigonometric function now again this doesn't decay to zero at plus and minus infinity but there are tricks you can play so the trick most common is to multiply this by a window function W where basically W is 1 in some window and 0 everywhere else, so now if I multiply W by my sine or cosine function, it has this nice property and then I can take the limit a As this window gets infinitely large, that's one way to Fourier transform these signals, but again it's a little annoying, so what I'm going to show you is how the Laplace transform is basically a one-sided Fourier transform weighted for these nasty ones. functions.

Well, that's all I'm going to show you today and then we'll use it later. Well, you know, this is all because of Pierre-Simon Laplace, one of the great mathematicians, so the solution and I make sure that you really can. write where you can see, so the solution is let's call these our little functions f of T ok, these little functions f of T, so our solution is to multiply f of t by some very stable e at least gamma t and at least gamma t, that is an exponential function e at least gamma t so that f of t e at least gamma T goes to 0 when T goes to positive infinity, ok, only when T goes to positive infinity, the first thing we are going to do is the solution: We're going to take our misbehaving function f and we're going to multiply it by a decreasing exponential function, so we're going to multiply it by a decreasing exponential so that when they are multiplied together it goes to 0 because when T goes to positive infinity now you might be thinking, well , we solved this problem in the positive direction T infinity, but now my function could explode at T negative infinity T equals negative infinity, so not only do we multiply by e minus gamma T, we also multiply with our useful Heaviside function H of T ok , now what we have is we're going to define this big function, the big f of T is going to be equal to some F times e minus gamma T times Heaviside of T, okay, that's what we're going to do is take our misbehaving little F of T and let's multiply it by a sufficiently stable exponential e at least gamma T, where if I multiply them it goes to 0 at infinity t and my Heaviside function so what I get is something that is zero for T less than zero that drives this thing that explodes at negative infinity and is equal to F of T and at least gamma T for T greater than or equal to zero well, so this this is it, we take our misbehaving function, the we multiply by a stable exponential and a Heaviside function and now we're going to Fourier transform the big F, so the Laplace transform of the small F is the Fourier transform of the big F, good and I'm going to write this here.

In fact, I'm going to frame it, so I'm going to write my Laplace transform pair just like we wrote our Fourier transform pair, so we'll have f of T. equals some Laplace transform, it's the inverse Laplace transform and the bar F of s will be equal to my Laplace transform, okay, so I'm going to save this and we'll fill them in later so the Fourier transform of big F is we're going to call that big f hat and it's going to be a function of Omega and that will be equal to the integral from minus infinity to infinity of large f of T e at least I Omega T DT this is what We always do when we do the Fourier transform, that is the Fourier transform, a large F and remember that I can transform the large Fourier F because I have multiplied it by the stable Gaussian function and the Heaviside function, so it is zero at plus or minus infinity, so I can transform you into large.

Okay and now what I'm going to do is substitute into my formula here this big F. The first thing I'm going to notice is that this Heaviside function is 0 for all T less than 0. so I can change the limits of the integral of my integral of instead of minus infinity to infinity, it's from 0 to infinity, so now this is from 0 to infinity and I can remove my Heaviside function from the little F of T e to minus gamma t e to minus I Omega T DT ok, so I changed my integration limits because of my heavy lateral function, so instead of negative infinity to infinity, I'm doing 0 to infinity and then what I'm going to do is group these exponentials are here and I'm going to say this is equal to the integral from 0 to infinity small F of T e at least gamma plus I Omega T DT and the last thing I'm going to do is say that gamma plus I Omega is my Laplace variable s, which is a pretty bad parenthesis, so I'm going to say that s is equal a gamma plus I Omega, that's my Laplace variable, so it's equal to the integral 0 to infinity f of t e to the S T DT and that's my Fourier Laplace transform. the transform of large F is my Laplace transform of the small F that is the definition of the Laplace transform is the Laplace transform the Laplace transform is the Fourier transform of a one-sided weighted function f is a one-sided expected transform in the which I think is like a political Fourier transform, okay, so I'm literally going to say it's defined, this is the f-bar of s, this is how I'm defining the f-bar of s, sorry, it got cut off a little bit. bar f of s is equal to integral from 0 to infinity of the small f of T e at least s T DT which is the definition of a Laplace transform and now the inverse Laplace transform is just the inverse Fourier transform of this large f of Omega, okay, that's what What I'm going to do now and maybe make a different color here, so now what we're going to do is say that the big f of T is the inverse Laplace transform of this and , if I remember correctly, it is 1 over 2 pi. negative integral from infinity to infinity of F that Omega e al now instead of minus I Omega T is plus I Omega T and we are integrating with respect to D Omega ok well and this f of T was small F multiplied by e at least gamma I remember I want a little bit of F of T for my inverse Laplace transform, so I'm going to take this and multiply both sides by e a gamma plus T, so e a gamma plus t e a gamma plus T and that will give me the little f of T that I want , so the little F of T is equal to this weighted inverse Fourier transform of the big F Omega and now I'm going to start working on what the mass looks like here, okay, so this is equal and remember that F hat Omega is the same as the f bar of s, these are the Laplace transform, the small F bar is the same as the big f bar and therefore this will be equal to 1 over 2 pi integral minus infinity to infinity.

I'm just going to change this and call this bar f of s and now my e to the gamma t multiplied by e to I Omega T is e to the gamma plus I Omega T D Omega okay, you'll recognize that this is our handy Laplace variable Now all I have to do is to change this D Omega and the limits of my integration to d s and I'm going to do that right now for you, so if I look at D s, remember that gamma is just a constant. is a constant that is large enough to go to zero, so gamma is a constant, so ds is I multiplied by D Omega, okay and that means D Omega is just 1 over I DS, so that I'm just going to change my D Omega. for 1 over I D is fine, so I'm going to draw my eye here on my coefficient 1 over 2 pi.

I think I'm going to run out of space here, so I'm going to do it down here, so this is 1 over 2 pi I integral of f bar of s e to positive s TDS ok and here's the last thing, the last thing is that if Omega passed from minus infinity to plus infinity if Omega went from minus infinity to plus infinity then s goes from gamma minus I infinity gamma minus i infinity to gamma plus I infinity ok so my integration limits changed and that's the inverse Laplace transform, that is all there is, so now F of T is just 1 over 2 pi 2 pi integral gamma minus I infinity to gamma plus I infinity F bar of s e to the positive st D s and this should look a lot like the parrot Fourier transform because It's a Fourier transform here the Laplace transform pair if I have some time function F of T u behaves badly.

I can take its Laplace transform this way and if I have the Laplace transform F Bar of s I can invert the Laplace transform and get back my function f of T, my original function and this works for nasty misbehaving functions that you might not normally we take the Fourier transform from ok, so let's take a step back, the Laplace transform is a generalized Fourier transform for misbehaving functions, so instead of directly Fourier transforming those misbehaving functions, which What we do is multiply them by a stable exponential so that they decay to 0 and a Heaviside function so that they do not explode at negative infinity and then we Fourier transform that product, so it is one-sided due to the weighted Heaviside function due to the Fourier e at least gamma t. so it's a one-sided expected transformation for misbehaving functions, that's all the little applause and it's extremely useful because many of the solutions from PD ES and OD es and control theory are more like this and this and this then our well-behaved functions where we can easily Fourier transform, so in the next lecture what I will do is explain to you some of the properties of the Laplace transform.

It inherits most of the same properties as the Fourier transform. transforms, for example, how you transform derivatives or convolutions and we will use those propertiesof the Laplace transform to simplify our PDEs to the algebraic Odie equations and we will also use this a lot in the control theory training camp. Alright. Thank you