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Instructor (Brad Osgood): All right. So today, I wanna continue our study and begin a real – serious mathematical study of the question of periodicity. Remember that we are essentially identifying the subject of Fourier series with the study – with the mathematical study of periodicity. And last time I went on, at some length, about the virtues of periodicity, about the ubiquitous nature of periodic functions – periodic phenomena in the physical world, and also in the mathematical world. And we made a distinction, perhaps a little bit artificial but sometimes helpful, between periodicity in time and periodicity in space. Those sort of two phenomena seem to be, or often come to you in different forms, and it’s sometimes useful in your own head to sort of ask yourself which kind of periodicity are you looking at? But in all cases actually, periodicity is associated with the idea of symmetry. That’s the topic that will come up from time to time, and if I don’t mention it explicitly, as with many other things in this course, it’s one of the things that you should learn to sort of react to or think about yourself – see what aspects of symmetry are coming up in the problem, how does a particular problem fit into a more general context because, as I’ve said before and will say it again, one of the wonderful things about this subject is the way it all hangs together and how it can be applied in so many different ways. All right. If you understand the general framework and put yourself – and orient yourself in a certain way, using the ideas and the techniques of the class, you’ll really find how remarkably applicable they can be. Okay. So I said – as I said last time – as we finished up last time, when we’re sort of still just crawling our way out of junior high, a mathematical course of periodicity is possible because there are very simple mathematical functions that exhibit periodic behavior, namely the sine and the cosine. But that’s also the problem because periodic phenomena can be very general and very complicated, and the sine and the cosine are so simple. So how can you really expect to use the sine and the cosine to model very general periodic phenomena? And that’s really the question I want to address today. So how could we use such simple functions – sine of t and cosine of t – to model complex periodic phenomena? Now, first, the general remark is, how high should we aim here? I mean, how general can we expect this to be? So how general? All right. That’s really the fundamental question here. And in answering that question, led both scientists and mathematicians very far from the original area that they were investigating. Let me say – well, let me say right now, pretty general, all right. And we’ll see exactly how general – I’ll try to make that more precise as we develop a little bit more of the terminology that really – that will apply and allow us to get more careful statements. But we’re really aiming quite high here, all right? And we’re really hoping to apply these ideas in quite general circumstances. Now, not all phenomena are periodic. All right. And even in the case of periodic phenomena, it may not be a realistic assumption. I think that’s important to realize here what the limits may be, or how far the limits can be pushed. So not all phenomena, naturally, although many are, and many interesting ones are periodic. And even periodic phenomena, in some sense, you’re making an assumption there that is not really physically realizable.