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• Well, let's get started. Captain Tsubasa J Psx Roms. The topic for today is -- Sorry. For today and the next two lectures, we are going to be studying Fourier series.

Today will be an introduction explaining what they are. And, I calculate them, but I thought before we do that I ought to least give a couple minutes oversight of why and where we're going with them, and why they're coming into the course at this place at all. So, the situation up to now is that we've been trying to solve equations of the form y double prime plus a y prime, constant coefficient second-order equations, and the f of t was the input. So, we are considering inhomogeneous equations.

This is the input. And so far, the response, then, is the solution equals the corresponding solution, y of t, maybe with some given initial conditions to pick out a special one we call the response, the response to that particular input. And now, over the last few days, the inputs have been, however, extremely special. Can Non Smart Phones Get Hacked Email.

Bluetooth A2dp Driver For Windows 7 64 Bit. For input, the basic input has been an exponential, or sines and cosines. And, the trouble is that we learn how to solve those. But the point is that those seem extremely special. Now, the point of Fourier series is to show you that they are not as special as they look.

The reason is that, let's put it this way, that any reasonable f of t which is periodic, it doesn't have to be even very reasonable. It can be somewhat discontinuous, although not terribly discontinuous, which is periodic with period, maybe not the minimal period, but some period two pi. Of course, sine t and cosine t have the exact period two pi, but if I change the frequency to an integer frequency like sine 2t or sine 26 t, two pie would still be a period, although would not be the period.

The period would be shorter. The point is, such a thing can always be represented as an infinite sum of sines and cosines.

So, it's going to look like this. There's a constant term you have to put out front. And then, the rest, instead of writing, it's rather long to write unless you use summation notation. So, it's a sum from n equal one to infinity integer values of n, in other words, of a sine and a cosine. It's customary to put the cosine first, and with the frequency, the n indicates the frequency of the thing. And, the bn is sine nt.

Now, why does that solve the problem of general inputs for periodic functions, at least if the period is two pi or some fraction of it? Well, you could think of it this way. I'll make a little table. I'll make a little table. Let's look at, let's put over here the input, and here, I'll put the response. Okay, suppose the input is the function sine nt. Well, in other words, if you just solve the problem, you put a sine nt here, you know how to get the answer, find a particular solution, in other words.