
Fourier Series 9

Lecture1.1

Lecture1.2

Lecture1.3

Lecture1.4

Lecture1.5

Lecture1.6

Lecture1.7

Lecture1.8

Lecture1.9

Fourier Series Part 5
This is a recap of the [last] Tutorial which is up was a monster about 27 minutes long I’m sorry for that But you know for analysis is just a monster. [so] I would still encourage you to go through it This is just a quick recap, but I would still encourage you to go through that monster because I believe it is well done [I] looked at it checked it, and it’s solid so I would encourage you [to] go through it But let’s take a look at a recap So here’s the general idea behind fourier analysis you given a function like this some function That [is] stepwise the thing is it comes up it shoots over to come straight [down] And so it’s a a stepwise function that you would have a difficult time [modelling] mathematically joseph fourier genius Mathematicians from France came up with this idea that if you if you can that any function can be represented by a constant plus a combination of cosines and sines Given that you find these coefficients right here And he derived these coefficients to be this this and this right well this coefficient these are coefficients, and this is the constant so he derived this constant and then each of these are the coefficients, okay, so the basic idea is to find these three things and then plug into this thing okay, so First thing is that we notice that this is an odd [function] right because it comes down over like this Because this is an odd [function]. We model odd functions with odd functions So we use odd functions to model odd functions so the sine the sinusoidal function is an odd function the cosine [function] is an even function, so All of these cosine terms are going to be going right? So you’re going to only going to need to solve for the b’s so on an exam? You need to make note right away, okay? Is the Piecewise function this crazy bizarre thing you know that I’m being asked to model is this thing even or is this thing odd? Once you determine that then you determine? Whether then you can determine which one which one of these you’re going to be using right? Okay that will save you tons of And that will happen I [hear] to prevent that’s how that’s all professors do it because it takes too much time Otherwise they have to do that because you can’t finish it in an hour’s time okay, so You go through and you find that you find for this one. We were just finding the bees So we show that the [aides] were zero we found the bees by plugging into this formula, L is equal to Pi So we go through we find B1 We go through we find B2 B3 before and so on and you [can] look at the tutorial for the guts of that, okay? and then we noticed eventually that B2 that all the even B’s all the B2 B4 B6 be a they were all equal to zero so we all had all we had to do was find the odd B’s and Eventually we noticed the pattern we notice that For the three well for the one it was [for] C over [PI] and then for the three it was For C over three Pi and then for the five it was for C over five [pi] so it was always 1 over N times for C over Pi Right so now you can just write [down] a bunch of terms you could write this here if you look at this final thing here if You come to here you could just write for C over Pi times sine of x for C over three Pi times sine of 3x? plus + 4 7 5 pi plus 2 sine of 5x plus 4 C over 7 5 times sine of 7 x [plus] 4 C over 9 PI times the sine of 9 [x] + 4 C over 11 Pi times the sine of 11 x + 4 C over 13 Pi times the sine of 13 x + 4 C over 15 Pi Times the sine of 15 x and so on and you can write down a huge series to model this thing really closely and you Could do it. Just like that. Okay next thing use your calculator So you can set up integrals with an evaluate sign So you can set it up to [where] you have 0 2 pI cosine Q times x? Dx and then put a little Evaluation bar and then on the [outside] the evaluation bar put q equals and then you could put you know Cubicles, then you put a 1 in there and then you bring it back into the editor and you change it to a 2 and Then you bring it back in the [editor] and change it to a three and so you can do that and I did that just to verify [that] all the cosine terms would come out to be equal to zero I just went through and Did those and you could do it for the sign also? So this is a very powerful tool [to] use to make to keep you from making Careless errors right, okay? This is just a recap [of] the previous tutorial and step all the way and give you a clear screenshot