TheFourierTransformAndItsApplications-Lecture04 Instructor (Brad Osgood):We’re on the air. Okay. A few quick announcements. First of all, the second problem set is posted on the website. Actually, I posted it last evening. So for those of you that are very eager and check on the website all the time, it was there. And secondly, the TAs are beginning their office hours this week, today in fact; is that right? Okay, so if you have any questions for them, they will be available to help. All right, anything on anybody’s mind, any questions about anything? Student:Um. Instructor (Brad Osgood):Yeah. Student:[Inaudible]. Instructor (Brad Osgood):Anybody else have any issues with the online lectures? I don’t know, I haven’t – I'm afraid to look at myself, so I don’t know what they’re like. Student:I was [Inaudible]. Student:Nothing happens. Instructor (Brad Osgood):Nothing happens when you click on it? Student:[Inaudible]. Instructor (Brad Osgood):It’s a little trick we like to play on people. Student:[Inaudible] which browser you are using, so in the Mac [inaudible], should be [inaudible]. Instructor (Brad Osgood):So the question may be which browser you’re using. I honestly don’t know; I’ve never tried to do it before [inaudible]. Student:If you’re using a Mac, you have to use Safari; it doesn’t work on anything else. Instructor (Brad Osgood):It doesn’t work on anything else, except – use the Mac, the word from over there is you have to use Safari, which is the one that comes with it. And I don’t know about other ones. Anybody else have issues with this? I can find out and I can post an announcement, I suppose. But I haven’t heard, actually I haven’t tried it, so I don’t know that the – how do they look, the lectures?Student:[Inaudible]. Instructor (Brad Osgood):Great. Thank you; that was the right answer. Anything else? All right, so I’ll check into it, but try that on the Mac, try Safari or try other browsers. Any problem with PCs? Student:They work fine. Instructor (Brad Osgood):PCs work fine; okay. Don’t, don’t – I don’t want to see. All right, anything else? All right, so today I have two things in mind today. I want to wrap up our discussion of some of the theoretical aspects of Fourier series. We’re skimming the surface on this a little bit, and it really, you know, kind of kills me because it’s such wonderful material and it really is important in its own way. But as I’ve said before and now you’ll hear me say again, the subject is so rich and so diverse that sometimes you just have to, you can’t go into any – if you went into any one topic, you could easily spend most of the quarter on it and it would be worthwhile, but that would mean we wouldn’t do other things which are equally worthwhile. And so it’s always a constant trade-off. It’s always a question of which choices to make. So again, there are more details in the notes than I’ve been able to do in class, and will be able to do in class, but I do want to say a few more things about it today. That’s one thing. And the second thing is I want to talk about an application to heat flow that’s a very important application historically, certainly and it also points the way to other things that we will be talking about quite a bit as the course progresses. All right, so let me wrap up and again, some of the sort of the theoretical size of things. And I’ll remind what the issue is that we’re studying, and so this is our Fourier series fine, all right? Last time we talked about the problem in trying to make sense out of infinite sums, infinite Fourier series, and the important thing to realize is that that’s not by no means the exception, all right? We want to make sense of infinite sums of complex exponentials sum from K equals Minus Infinity, Infinity, cK, either the 2 pi, KT. I'm thinking of these things as Fourier coefficients, but the problem is general. How do you make sense of such an infinite sum? And the tricky thing about it is that if you think in terms of sines and cosines, these functions are oscillating. All right, everything here in sight is a complex number and complex functions, but think in terms of the real functions, sines and cosines where they’ oscillating between positive and negative, so for this thing to converge, there’s got to be some sort of conspiracy of cancellations that making it work.Of course, the size of the coefficients is going to play a role as it always does when you study issues of convergents. But it’s more than that because the function is bopping around from positive to negative, see, all right and that makes it trickier to do. That makes it trickier to study. And again, realize that this is by no means the exception, and so in particular if F of T again is periodic, period 1, we want to write with some confidence that it’s equal to its Fourier series. We want to write with some confidence, at least we want to know what we’re talking about, that F of T, say is equal to its Fourier series going from minus [inaudible] 2 p i KT, and again, it’s really if you want to deal with any degree of generality, it’s going to be the rule rather than the exception that you’ll have an infinite sum because any small lack of smoothness in the function or in any of its derivatives is gonna force an infinite number of terms. A finite number of terms, a finite of trigometric sum will be infinitely smooth. The function and all its derivatives will be infinitely differentiatable, so if there’s any discontinuity in any derivative you can’t have a finite sum. So any lack of smoothness forces an infinite sum, again, so it’s not because the method is trumpeted as being so general, you have to face the fact that you’re dealing with an infinite number of terms here, all right? Now, by the way, I don’t mean to say that all the terms are necessarily non-zero, that all the coefficients are necessarily non-zero. That’s not true. Some of the terms may be zero. For example, when you have certain symmetries, the even coefficients may be zero or the odd coefficients may be zero and in special cases, or a finite number may be zero or a block of them may be zero. You don’t know exactly what’s gonna happen. But all I'm saying is you can’t resort to only a finite sum if there’s any lack of smoothness in there. All right, so again, that’s the issue. Yeah. Student:[Inaudible].