Lec 16 MIT 18.03 Differential Equations, Spring 2006
Okay, that's, so to speak, the text for today. The Fourier series, and the Fourier expansion for f of t, so f of t, if it looks like this should be periodic, and two pi should be a period. Sometimes people rather sloppily say periodic with period two pi, but that's a little ambiguous. So, this period could also be pi or a half pi or something like that as well. The an's and bn's are calculated according to these formulas. Now, we're going to need in just a minute a consequence of those formulas, which, it's not subtle, but because there are formulas for an and bn, it follows that once you know f of t, the an's and bn's are determined. Or, to put it another way, a function cannot have two different Fourier series. Or, to put it yet another way, if f of t, if two functions are equal, you'll see why I write it in this rather peculiar form. Then, the Fourier series for f is the same as the Fourier series for g. And, the reason is because if...