
Fourier Series 9

Lecture1.1

Lecture1.2

Lecture1.3

Lecture1.4

Lecture1.5

Lecture1.6

Lecture1.7

Lecture1.8

Lecture1.9

Fourier Series Part 1
Joseph Fourier developed a method for modeling any function with a combination of sine and cosine functions. You can graph this with your calculator easily and watch the modeling in action. Make sure you’re in radian mode and let c=1:
f(x) = 4/(pi)*sin(x) + 4/(3pi)*sin(3x) + 4/(5pi)*sin(5x) + 4/(7pi)*sin(7x) + 4/(9pi)*sin(9x) + 4/(11pi)*sin(11x)
What if we were charged with the task of modelling this function mathematically? So we see that this function is periodic and that it’s a step function So that [it] comes up over down over up Over right, and so we see that It’s it is in fact periodiC okay? so we might hypothesize that we could use sinusoidal functions to model this function because let’s take a look at some [sinusoidal] function here is the sine function and the cosine [function] and We see that these are periodiC functions So maybe we can use these to mathematically model this and we would be right in making that assumption so Joseph Fourier in the early 1800s [hypothesized] just that and then proved it, so let’s take a look at what he proposed and then later proved so he proposed that this function could be approximate by some constant plus a sum of cosine and sine functions a Combination of cosine and sine functions and that any function could be modelled with this Given that you could find the coefficients these this a sub n And this b Sub N to match it up right to make things fit, okay? and also this constant here a sub 0 okay, so We look at these and let’s look at the sine function. Let’s look at this cosine function Which one of these do we think is going to do a better job of modelling this function which one? Do you think will do a better job? Well, I would guess that the sinusoidal function would do a better job the sine function that is approximately Matches The period well it exactly matches [the] period look this one has come from negative Pi and negative pi it’s coming down and over at negative Pi [this] is coming down and over and Then from 0 this is 0 to pi it’s coming up and over and then this is coming up and over So if we superimpose this on top [of] this it would do like this So this sign function is more is a better approximation To this then this cosine function is the cosine is off, right? It’s out of phase with the function that we’re trying to model. So if we were using these terms here in Our mathematical model that was the mathematical model that we used to model this These cosine terms would be working against us So we conclude the following when we use for Modeling odd functions, let’s use the sine terms the sine function and For modelling even functions let’s use the cosine function makes sense Now there will be some functions remember I said any function So there’d be some functions where you will use a combination of cosine and sine terms but for this one because it’s strictly a an odd function you will only use the sine terms to model this function and In the next tutorial in part two of this series, we will show that that’s true. Okay here. We go So let’s take a look the next the real hard part the brilliance of Joseph Fourier was identifying what these constants would be and Here, they are So Joseph Fourier and the early 1800s said what if we let the period be equal to 2l? then these things would be true okay, so The proof is in your book we will just do we’re going to do an application and zip through this kind of quickly But the application I’ve done this I’ve been in your place I’ve learned this before and the application is where you really see it come together when you graph it And you see how the what will happen when you graph it is the function will come up You’ll watch it on your calculator It will do it and they’ll come [up], and I’ll do this number and then when it gets to here It’ll come it’ll come it’ll zip down like this And I’ll do this like that so and The more terms you add if you added an infinite amount [of] terms, then it would match this thing exactly But of [course] no one can add an infinite [amount] terms because infinity is not a reachable thing but we can add a lot and so these these squiggly lines will get the humps will get smaller and smaller and smaller until it looks like you can you can’t see it with the naked eye you have to zoom down you know with your With your zoom function your calculated at the zoom down really tight to see the wiggling all right the squiggling lines [ok] so you see that this this for a 0 it goes from negative L to L They’re all integrating from Negative L to L so half the period in right half the period to the positive of the other period But notice Of the same period It’s all one period notice something that [this] thing goes from negative L to L so that’s one, Wavelength isn’t it that’s one period right Because this one goes the period it goes from negative Pi one cycles is this this is one cycle [boomboomboomBoomboom] that’s one cycle, right? Negative [Pi] [2] [pi] so what’s the total period And what we moves over by PI and then pI again? so the period is 2 Pi So the period is equal to 2 Pi and [so] solving for l. L is equal to Pi So we would integrate from negative Pi to Pi and then this f of x here is The function itself that you’re trying to model, but notice that this function We can’t just integrate we can’t it’s not a continuous function [as] a step function so we can’t integrate continuously from here to here So we have to break it up into [pieces] so I’m going to write f of x I’m going to write a zero right here Step out the way here a zero Will be equal to [one] over L Alice, L is equal to Pi 1 over Pi 1 over 2 Pi Yep, two times ll. [s] pI okay times Open Bracket the integral from negative Pi to PI and We’re going to one note pardon me. We’re going to go from negative Pi to [zero] we’re going to break it up and then f of x From here to here f of x is equal to what negative C So we put f of x in here. This is negative C Dx plus from 0 to Pi and Then from 0 to pi look the function is C from 0 to pi it’s positive C [ok] and if we do this, we will see that a 0 is equal to 0 Right because look at it You have this area here. Which [is] the negative area and you have the same area which is positive So they cancel and a 0 is going to be 0 for this example But this is just to show you that how we’re going to break this up and We’re going to do this trick here. We’re going to do this trick for a sub n and b Sub N and a sitin will have a a 1 a 2 a 3 a 4 and 4 B Will have b 1 b 2 b 3 b 4 but question what do you think all the a’s are going to be equal to? 0 why because This is just this is [not] helping us this is not a good model this guy is the one that’s matching up closest this guy is working against this and joseph [fortier] developed the model or a mathematical construct that is very clever and Selfcorrecting if you will because all of these terms come out to be 0 and we’ll show [that] in the next tutorial I’m going to step aside and give you a clear screenshot part 2 is coming