PHY3600 G2 FOURIER SERIES

[00:00:00]All right, assalamualaikum and very good evening. And good night everyone to tomorrow's final exam. So, here is the Fourier series. And this is really important.

[00:00:12]You try to study last minute here.

[00:00:16]So, we have here um function and then one.

[00:00:23]Okay? And then the interval is from zero to pi over two. And then the second function here is zero. And the interval here from the negative pi to pi. So, basically um this is one type of example of question and maybe you can be done in the final exam. So, a laser source generate optical pulse described by this function.

[00:00:48]So, you have to expand the function in a Fourier series to provide at least three cosine terms and three sine term.

[00:00:58]So, what you have to do here um step by step.

[00:01:03]So, to expand this function into the Fourier series, so we use the standard Fourier series represented um for a periodic function with the period of T is equal to 2 pi. Okay?

[00:01:17]Since the the interval is from negative pi to um pi. So, here is the standard now. The A0 over 2 plus the summation of AN which is a cosine term and then this is a sine term.

[00:01:34]Okay? So, as I mentioned before, you have to choose what I'm sorry, you have to find A0 AN and BN. So, you have to find this A0 um the cosine coefficient and then this one is sine coefficient.

[00:01:54]Okay? So, first of all, calculate the A0. Okay? So, this is a general formula.

[00:01:59]You can also write in the cheat sheet. Okay.

[00:02:05]Okay.

[00:02:06]And then the um so this is the uh general formula of to find the A naught.

[00:02:13]And here the function of T the first function here is one and zero. So it is easy because we have only one. So zero So the function of T here is one.

[00:02:28]Okay.

[00:02:29]And then the limit here from zero to uh pi over two.

[00:02:36]And you have to integrate.

[00:02:38]Okay, one with respect to T. So integration of one with respect to T you are going to have that.

[00:02:45]T.

[00:02:46]Okay, the interval is from zero to two pi over two. And then you plug in the value of T. So what is T? T is pi over two minus zero. So pi over two minus zero. And then you are going to have that. You will find you will get A naught is equal to one over two.

[00:03:06]Okay, one over two.

[00:03:08]And then you see here is the first term is A naught over two. So A naught is one over two divided by two. Final answer will be one over four. To check.

[00:03:19]Okay, this is the sign. So to find the A naught. So we have that and then we have to find the second coefficient which is cosine coefficient which is A N.

[00:03:29]So and then next is find to to find the A N. So this is the general formula of A N. And then again the F T here is only one. Okay, so one here. So the integration of uh cosine and then the integral zero to pi over two. The integration of cosine with respect to N T with respect to D T.

[00:03:49]Cosine N T with respect to D T. You are going to have that. Sine N T over N.

[00:03:54]Okay. the interval is pi over two.

[00:03:57]As zero to pi over two. And then if you calculate this one Okay, simplify it.

[00:04:05]I will put pi over two in the T and then put the pi over two inside the T.

[00:04:11]Okay, you will get this is the final answer of AN.

[00:04:15]Okay, so check.

[00:04:19]And then what you have to do is Okay.

[00:04:24]Right.

[00:04:25]Yeah, you have The next step is find the first few non-zero, which is find the first three term.

[00:04:35]Sorry.

[00:04:36]First three term. Find the three term of the AN.

[00:04:41]What you have to do is put the value of n from one, two, and three until you get the first three.

[00:04:50]Um term. Okay.

[00:04:52]And if you put Sorry.

[00:04:57]If you put n equal to one, I will put one in the n here.

[00:05:02]I will put A1.

[00:05:03]And then the first term is one over pi.

[00:05:06]And then you if you put n is equal to which is the A2 for the n.

[00:05:12]And you will put the n two and put the n in the n.

[00:05:17]Okay, this is the A2.

[00:05:19]So AN is two pi over two divided by n pi is just two times pi. Two times pi. And then you will get zero. So it's not included because we have to find the non-zero term. Okay. And then if you put n is equal to equal to three, you will get one over three pi. So it is included.

[00:05:42]This is not.

[00:05:44]And then if you put n is to to four you will get the answer is zero. It's not included. Now, if you put n is equal to five, five as n is equal to five, okay, the final answer will be 1 over 5 pi. So, this is This are three first three terms for um What do we call this? The sign coefficient.

[00:06:12]Okay.

[00:06:13]1 2 3. Okay. And then, the next step is to find the bn. So, the general formula for bn is 1 over pi uh interval of -pi over 2 pi ft function sin nt over dt.

[00:06:34]Again, if you um you integrate sin, you will get negative cosine nt over n, and you have to put the interval over pi over two in the into the t uh with respect to t, and the final answer of bn, you will get this one.

[00:06:55]Sorry.

[00:06:58]This one.

[00:06:59]1 over minus cos 2 pi over 2 divided by n pi.

[00:07:06]Next step, you can try three first three terms. So, the question is to try three terms.

[00:07:11]Okay.

[00:07:12]Put n equal to one. Start from n is equal to one.

[00:07:17]And you will get 1 over pi. If you put n is equal to two, you will get 1 over pi as well. And then, if you put n is equal to three, you will get 1 over 3 pi. So, these are uh first three terms.

[00:07:35]Okay.

[00:07:36]And the first three terms for the sign coefficient for an, okay AN and then for BN these are the first three terms which is the cosine term.

[00:07:52]Next you have to do is you put all the A naught AN BN into the Fourier series here.

[00:08:11]And then this first three term and then this is first three term for the sine term.

[00:08:23]AN upper core first first three term and then the first three term for sine coefficient. So that is uh step by step how to find the how to expand the Fourier series. So it's really important.

[00:08:39]Uh try to study I will give you last minute so that you will you all depend on me waiting for me so that you will get last minute so that you fresh remember for tomorrow. Okay, see you tomorrow. Bye-bye.