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3Joseph Joseph Korean pronunciation is correct. Yes. He's a very powerful this is as well as he has. 
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5So he has Things are pretty serious for a Transformer later, but it's not covering this and heat conduction. You probably have heard about you through all his so-called fully know of for heat conduction. Basically what he says is the following if you have some food of metal necessary one side is swarmed the other size hole and then There will be heat energy flowing from the Hawks. What does I threw the cooler side? And according to this relationship? There is a fair amount of heat energy is proportional days of Eternity constants. That's the cross-sectional area of the conductor. Delta T is the temperature difference. And then live here. And also, of course the amount of energy through across this conductor depends upon how much time because we are talking about. This is kept under constant temperature that is bottle cap on the top. Now. This is kind of one dimensional description of the heat conduction properties. You would like to know What happens if you have a three dimensional set and in differential forms and this you can bring the tea to us the other side? All right, or they are afraid and that is the heat flow rate per unit time and it is proportional to that and if it is hit troll is going in three-dimensional and this is tested radium in the well one dimension. But in the three-dimensional situation will probably have a great right. This is the gradient Direction because the heat stroke is always from hot to cold. So this is an air-conditioned place. The radian is always from look at the temperature field is from low temperature pointing to us to a higher temperature this video therefore expect. Of course. It depends on how you measure the production of that is not very essential now seems that the Hope chapter here is the only thing I have to see related to physics or any physical example from what's remaining is entirely formula and mathematic with very to figure. Okay. This is one In engineering be many situations, you will encounter periodic signals. I supposed to get think of that circuits. Sometimes they oxidate and then you have some sort of periodic signals even engines and it is running at a constant speed. Wow, and you will see Kristen going round and round and you have periodic table. So this if you have a periodic signal you can always Has be broken down into component. Opponents periodic components so fucking show on top here. The top figure is a what if you how many calories you have time to figure out that is Security on it. But in fact it is something that can be created by adding a few sideway sine wave or cosine wave. These are the fundamental components of periodic table. So we want to press down. General periodic signal in terms of components. So this isn't a number of components can be infinitely many not justify number of all right, you can be Yes countable infinite number of components. Okay. So what are periodic functions? I believe you have heard about your periodic functions in the functions graph, but it's simply a graph with repeat itself one after the other after certain fixed purest form. So we say that a function f of T is periodic if there is a Number three subsets have t plus capital t is equals to the same function, but l t plus Delta t means How is that relate with f of t f of T and F of t plus delta T was well, if you have a problem ever have t plus Delta t means Shifting the graph to which side left side right side. Sure thing the rock right I see is the number two team. I would shift the the graphs right horizontal, but whether shipping to the Earth or to the right? 
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7You try yourself shifting To the Left Right Moving the cross will move by if you're shipping by T unit and the whole graph will move by 2 units by adding T because you see when you put T equals to 0 now. It is another Point here. So you have to move that point to the left. That you get back the same function now this Is for all T. All right for all so we are looking at here is a function when it is periodic is essentially a function that is defined over the home kind of 40s. Now. The number T. Here is for Superior and sister function is also sometimes for T capital T. If you wrote it, unfortunately, he's smart. He's one of the T capital T. I say yes to use that description because they are the same length. So you don't keep your idiotic means that the function has a Puritan T capital T. It is stay pure it because you can make it t if you have a periodic function. F of t write this F of t plus 2 T is also true. All right is also true. It is also true that you have to have sweetie. So well Tootie is also a purity treaty is also very so that's why we just refer capital T as a purity know the how has that many applications and I have already mentioned that area. Yes. Well, of course here if you put in negative so much to work. Usually it is not accurate. It's not a concept that is that what is that? What is that? And then one is Deputy few. But what is T. What is technical team? I don't know to frequency because of negative frequency, right? We don't tend to associate with negative number. Of course, you are quite right that if you put a negative value here is still okay, right? Yeah. Well, we sit with positive from right. Let's see convention. That's it. No, yeah, that's what I have just explained. They are all here. So the smallest value that you can choose to make it. That is to make this work. Is It's so called fundamental here the smallest capital T that makes it work. To describe this property are responsive from the vendors have more sympathy the pyramids right when they say is the period increases in the fundamental Theory. So cosine T sine T. Wow. These are the Very basic functions that strike of the nation and cosine T sine T. They have the same fundamental period of two parts. Wow, that is the basic thing, but you can oxalate faster. By X A number life positive integer, right putting a positive integer because I can t scientist will be oscillation fast. So you can think of that. The fundamental here, it will become shorter which is equal to 2pi over. Usually this n value here is referred to the angular frequency angular frequency higher the angular frequency the higher the value of M and shorter period 2pi is always. I'm hearing a lot of these functions right? Although this is fundamental period is 2pi over n so if you have a fundamental period you multiply to any scalar integral multiple X any integer positive integer will give you also a purity of that function but the fundamental theorem of cosine NT int it's That what happened with your constant humiston hear. What I said is a function is periodic. It has no fundamental current that is only one way of looking at it. Yeah, because if you have a constant you can shift it to the left by any value. It's still the same constant. So it has no fundamental Theory. So for comes down for constant, you can have f t plus T equals to F of T equals to a constant the same constant for any. T greater than 0 all right for any T greater than 0 so that's why you don't have a different dimension, but it still periodic. 
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9How are you can't even look at it another way and saying that constant function has you can assign it to having any fundamental, right? So what's the difference between that and that? Yeah, these they have fundamental truth because you just want to refer to the fact that the fundamental Unity is a too high over only guys. It's similar to what we have here and they these are these frequencies and only right they are called harmonics so only If you this from the rental period and any multiple of Omega, they are for harmonic of this basic frequency. And again, they are for having tea has Superior because any Fela voted to any integer multiple of The fundamental period is also clear. All right. Any questions any questions? Okay. No question. That's very good. Now this is a property about To be any linear combination of two periodic function with the same good. Let's say F of T is a periodic function with period capital T and G C as well as any linear combination that is you multiplied by a scalar to one function and get to As a function multiplied you estate up on people as well. And you just add them up that is also a periodic functions, right? You can check brother easy, right? You can just find out that what makes t plus T. And then that is Alpha f t plus T. And then that's beta XI t plus capital T. And then this is f of T and then this is beta Chi of R t so that is H of T. So the proof is at almost. So that's why. If you sum these two quantities. Some of these Concepts All these Periodic function constant is a periodic function of any queries. You can think of it. That way this cosine T has to it to have this. Also has spiritual power this fundamental building is hard but 2 pi is also is theory and also all these for n equals to some integer similarly sign language only sign tells period 2pi so here the period is too high. The period is 2pi over 2 plus T is also pieces to pi over n, but they also more of them has. 
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11All of them 
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13Have your tea? To us Good to pop so all of them have period Tupac. Actually, this does not come directly from here because linear combinations technically is only applies to a final number of Eastern Syria, but it works. Okay. Now some of periodic function is not necessarily. Now we'll make me. Confused because I just said something or even me hating the population of your Iana functions will lead to a periodic function here. I said probably on its functions is necessarily periodic just because yes, 
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15Well, if you have a comment or you can always extend it multiply the type some scalar multiple rational numbers have a common fear, but fear what is indicators for you to be cautious? Is that if you have a function like this as sum of two periodic function, the first one here is the period is 2pi the period Is fun Okay. 
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19And they are both periodic Force if you process. How dysfunctional GT are using some software. They are function GT looks pretty nice periodic, but in fact, this is not periodic because you cannot Have balls Express the ratio of the period as a rational number. So what it means is that this you think of it decide when this thing hits Zero every too high this one gets zero every unit 1 unit. They would never meet then. We'll never need it after they stopped. Of course when T goes to 0 they start both of them start with 0. But so when T equals to 0 1 is something like that and the other Let's say this is 1 this is 1 this is 1 half and the other is too high. So too has some somewhere three something either one to and and They would never meet. At the same Point again. So that's something so carry on. All right, so be careful of when you add two functions periodic, but the Puritan society. It's not. Chorus puzzle correspond to a ratio of rational number Any questions about from graph? Okay. Yes, because what is hitting at the rec irrational course irrational that that one open seat at the irrational number for the other? It's a Russian. So you cannot get the perfect toy because you do rational number and and the greatest National and international numbers don't match here is 1/2. Three That's 1 2 pi 6 Pi so they will never hit the same point. Yes, because one always hits a rational point. The other always has a rational point so they would never 
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21Well, when you think you have to be careful because we you can have it of other tree long time, but when you talk about into the time, I don't know what Good because now you can have it goes up to a 10 billion or whatever then no matter how long you go in terms of the tx's you you never hear me. Say the same point zero. 
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23Okay. Okay. So now in the next section is to start to do some work for a series representation. As I said in the very beginning we want to be composed a periodic signal into its component and the number of components. Could be innocent. All right, because sometimes you need an infinite number of components in order to match a simple periodic function now because we know that based on what we have observed is not periodic function we can start with something. We assume that it is a 2 pi periodic. We assume that given This is a function with URI to path. But you say well I might periodic function is the pivot is 0.5 x 7 and what happens is that you stay on the X right you put a scaling on the axles. So any function? Periodic function you have you can always stay over the the T value so that It gives a period of two half after sale. Okay. So the question is can we write a few days Oleg function of your 2 pi into this foam? Why is this fault? Because this is all you can think of us Curie on extensions with the basic. Entities for these basic components of 2pi periodic functions became be a constant. It can be a cosine T. You can be consigned to T cosine 3T. They are all to five you reorder. So the sanctity they are also too high. Periodic, so we have here. All possible scientist Ian concept but this is actually you can think of that that also consists of the case n equals to 0 n equal to 0 is a constant right cosine 0 is what And we don't need to have p n equal to 0 here because science 0 is actually 0 so we only have this constant term which carries a loan so that it is related to the haps associated with the post - but you would definitely ask why 1/2? Why do we put 1/2 you? See that late. When you grow up very soon in a few minutes. Yes. Yes, of course you can use for your phone. To make it a comfortable exponential but we don't need here because it is a little bit easier to do when you've received for you. So here we have 1/2 perspective seems to be a bit awkward, but you will soon find out why we have this one factor. So the question is how First of all, I do is I say this periodic function with two Pi period equals to this state whether it's really always makes sense. The first thing you need to look at is that this equals if I write it this way then can only find these components A and B is constant for each other at this point. Not yet each other. So, how can I determine these decisions? Now we can come feel this school officials one by one, right? So how to do that. The first of all you can integrate what we have. Yeah. Integrate both sides from - title passed right over 1 over 1 period from minus pi to pass this. It's just that integral one here. If you don't write it explicitly and you integrate all these combine like terms. The first term here Is 2pi so the term here is 2 pi. And if you integrate cosine this term over one period What has easy I find Bullseye integration and you substitute lower and talk about give you 0 value. Wow. That's do it always zero. For any hand For n not for n equals to 1 2 and so on right? Then the other one here. Turns out to be exactly equal to 0 also well because he's assigned way cosine way you look at the area over the one Purity the positive area and negative parallels always give you a cancellation, right? So for example a sign. Consigned right? So the positive part. And the negative Parts. No, this is not. I've drawn a little bit too much. Okay, so is here? So that's minus Pi that's high. So hot here and there so that cancels out right that's local sign. That's cosine Theta cosine. Well if you have Cosine 2T. What happens is that you just compressed another side of you and not another period if it is Duty. So I love with a joy when you say just be so you said you have 0 here for any n 0 here for any n equal to 1 2 3 and so what this is 2 pi now, this is something that you should know because it is the function provided. Well, if you don't know the function, how can we find out is representation, so FLT is no so you can find out its value and then this is 2 pi so that a note. Becomes One over pi times the integral from minus pi to Pi of f of T DT. So we have found one. You see this is why we have the 1/2 for one reason you see so nights without the 2 pi here with kinds of impact to and this has a form which we will see. Closely related to The eighth and other 8 and 2/3 c 1 a 2 a 3 Oh, I have forgotten to take a break. All right, let's take a break. 
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27Programmers part of Pop basically, we don't need to integrate exactly from minus pi to Pi we can integrate over any field of the function. So for example, you can even change that. Say from 0 to 2 pi as well as its number one Purity. That's okay. You can integrate. Let's say l 2 pi plus tell if you don't. If you want to do a little bit more work to do that as well the sale. Save answer because even this what that means is that you shift by an amount. Hell No matter as long as you keep 2pi in between your shift to any Direction behind area about because that under the curve of the same so tensed, so it doesn't matter whether you integrate from minus 5 times. You can move it like this as long as it is too high. And what is above the expert the T axis and Below bit theaters would have the same. So cancel out event. Some facts would be useful because we need to compute a note a and and also be an now that's true for all cases. That's long as n em. They are integers and we only look at non negative integers, right? So 
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29Over 100 PSI and T and cos IM T the product of 2 cosine terms with possibly different integer. It will give you well either zero. If you don't have the same number here that is evident. They are don't say always heal you. Equals to M. And this help feeling is a possibility until right because you have cosine square and T in this case. You definitely be able to find out a non-negative. If you have Sian kaan sign makes always zero then how about sign comes some the product? Sometimes I intended as I'm not Also zero Harrison and the best if they have the same what? Not zero because you can see in 10 and equals 0 and definitely this would be 0 but n is equal but not equal to 0 and not equal to 0 together and you have time so you have to find a way to remember all that. Hope that it's not just remember all these fact. Well, of course you can prove I guess you can use that some angle formula to establish these friends. Now with these tools we can compute a end here for are we multiply that equation right this equation? We wrote down earlier. Before that sf4 t equals to have they not us some variables to 1 to infinity and to resign and t plus b and bargain put PSI and T right to think. So this time I multiply both sides by concise and T as is not yet feminine and I just integrate both sides over one period and we choose for Simplicity from minus pi to Pi. Now this time, of course, you need to find out its value if it can be found. If F of T is given this one now you remember this is a cosine m t is Is a term that is cosine. So over here is definitely if you are single man. This one you take the product because I am T and sine NT of the night inside empty to integrate that one depends upon whether and equals to M or know if n is equal to M you have time. Otherwise see you. So intentionally, there is only one know of this infinitely many times that it's not see you. That is when n happens people to act those term here because int scientist because it's a mixed sine or a cosine whatever the value of M and N will give you over a period of after integration be too soon. So that's always equal to 0. That's always equal to zero this one also equal to see that 1 is equals to M. Then is the only term that is not zero. So this is this term nonzero only when? And supposed to have and in that situation we have well, so we have to set n equal to M or m equals to n so that the only time that comes out is equal to part because all other times give you zero. So this time? Will be equal to pass. So f equals to n then this is okay. Right and You don't have that anymore because that is fully only one time that it's so this time it's here that time is tight icons am so we move that tie to the other side. Right and this formula works for n equals to 1 2 3 and so on so in one single this formula we can come Cute or the answer using one single for me. And the important things is that we call why we have 1/2 for a not recall that a not. We have come up with the formula which Is this one? Okay. Remember that and this can actually be recovered using this formula which include study goes to 0, right? So this formula here actually also hand. Provide the a zero time so you although we proved here for n equal to 1 2 3 but then if you packing 0 cosine 0 here is one fact precisely if you the a zero-turn so because of that one half Factor now we can use one single. Formula Don't include all the terms involving cosine and constant which should be the case because when n equal to 0 when n equal to 0 this stuff actually is a constant right? We should use have a common problem. Now 4 p.m. I suppose you can guess what type you can work that out in a similar way. I can just highlight some of these. 
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31Okay. So for some time I can multiply. Sigh int and here We'll assign case little closer, but I don't want to confuse you over here. But you you can work out the side and teeth like maybe over here you can see. I loved is my f of T sine m t and I have here a North Side empty and summation n equal to 1 a and cosine and sine Mt. + PM sign. And sign empty so in the using the same idea of integrating. From minus pi to Pi that also integrate from minus. I Phi minus I 2 I minus 5i right then you will see that is something that can computers. That's zero. Cosine and sine mixed together zero. Yeah. Half and and should not be equal to 0 n is definitely not equal to 0 but M you if you want. A different value, but if you have m equals to n then this term will give you Hi when and equals to F naught equal to 0 but it is definitely not right and start from 1 to Infinity. So in this case the value that you obtain is high and that is only one term that is non 0 when M 10 different 0 so you have BN X hi. Has zero plus zero equals to this thing F of T sine NT DT so you can work that out the end equals to this. So it seems that the hopefully serious topic has finished because you're now can find out a and at the end all the coefficients or the coefficients. So what is what we want to do in the following is perhaps you look at some of course very important. The other thing is not to look at properties of this way of representing a periodic function. So what investigation I have avoided to ask in the beginning of is whether this representation gives you the function f of T. I wrote it in the beginning F of T equals to this. All right. Supposed to have but is it true? Is it true for any point t? This is the question that I have avoided in the beginning. So perhaps we can look at it. So how we can summarize The essential thing of this chapter these coefficients a and bi products include also single they are called the Fourier coefficients. I mentioned also you can integrate from 0 to 2 pi that's also fun for any to high-purity interval. All right. So this is what you need to remember. This is for 2 pi periodic for functions of other periods. You have to scale to scale the tea. Texas so that it's it up to 2 pi. So something you need to listen to careful. When you encounter. The Purity density is not equal to 2i but that I believe there would be plenty of exercise questions in the tutorial that you can practice. All right. So what next? It's humble. That's the first exam. So it must be a simple we have this Square wave also seen in signals in an electric circuit. Actually, if you want to be a little bit even more precise according to the definition of this. Function T equals to 0 is 0 at T equals to 0 0. So you can highlight this if you want to this one as well. That is the good things of coming to lecture because you have a chance to correct it. Okay, if you want to but it is not essential now. This is a 2pi periodic truth. Right every 2 pi you shift it? The same route Okay, that's the graph. What do you seem fully a series that is you need to compute all the Fourier coefficients. So you may recall this is what we want to write. We want to find am a lot now. What is a n a and according to the formula is this? So here F of T has to over the Over the to pie unit from -5 and 5 from - Thai to Thai we have trouble with zero part of it equals 1. Right. So this is what so the integration from minus type to tie. You have part of it equals to one part of it equals to 0. So that's why you have it from zero to five. It was the one because the zero therefore minus x 0 correspond to a single value and this you can integrate easily because it just comes int and this is equal to 0. If n is not equal to 0, of course, you can't put an equal to 0 here so we can come up with all cases and equals to 1 2 3 4 5 and so a no term you have to evaluate separate right you have to evaluate it separately putting any goes to 0 and this is what you need to evaluate and this. I guess. - Titan 0 is 0 from 0 to Pi is 1 so this function this integral give you value. It was fun. But make sure you recall you remember that you have point from don't think so. We have computed all the AIDS. Now for the be you guys have built this using the formula integrate side and team. That n is not equal to 0 so you can have one - well the upper limit putting the Upper limit and the lower limit this is because it's an extra lower limit and equal to 0 is 1 so, I would expect that when you encounter Consignment Pi n is an integer, you know, that should be written in this form. So this allows you to have some simplification. So this is a number That oscillate between plus 1 minus 1 and the value of n if n is an even number it is possible. That is too high zero to pie for my event is a both integer that this is minus 1 So what actually with minus 1 you have something with with his power it was because one that would assume so when you have a and n as an even integer, then it is see if n is an odd feeling teacher it is equal to - what so 1 minus minus 1 is 2 right so you can substitute n equal to 2 K plus 1 with now K goes from 0 K goes from 0 to 1 progressed really positive Infinity. Okay. So now we have computed every and You can combine them everything. Wow. That's a figure that can happen. 
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34So F of T. This is what happened is the eighth half of a note they know his voice and has all these time because the The cosine term they are all equal to 0 except the constant a 0 Zan all of them equal to 0. So this cap Valley that has vanished p and you can compute something. They don't you have the value 1 so that this if you expand everything term by term your Fourier series looks And like this or in this form now this starts from Zeal because I have actually changed came to end right again. Just you see here. I have used to k n equals to 2 K. Of course, you can put K here that since the I change it back to em for doesn't matter if you understand so The proper free series With one two three and seven pounds. So with one time that is one half. That's the constant. So this has a sense of approximation. Just like Taylor sixth sense. If you in please the number of terms in this what would you say? You want to omit the times higher frequency and you see lower frequency terms as approximation and you can see by introducing higher and higher frequency term. What would happen in this case one term that is 1/2 pain is the static. Freak 
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36That is the static State and this is actually just one now. You can see why this is one because that term with only one time the constant term essentially tries to have rage the city. I have constant. So if you have a signal like the one Like that. What is the if I asked you to use a single constant to approximate that of course, you will take a test and that is what That is you use. This is 1/2. Okay, you use 1/2, too. Represent that sing If you use 1/2, if it's not good enough, I take one signs. So I take the first nonzero sign. But I suppose you can guess is would be this blue curve because we have one side way has a constant. So if x onto this Thompson sideway, right? One more time is three times equal to the Yellow Cup. And then seven times in total look up. So the more worried if you cut more and more terms you will have At least it looks like you have better and better approximation of the original function the original function f of t look something like this 0 here. And then one day let's see F of T that you have. Right half of 2 F of T looks something like this. And a zero here. Let's have a party so looks not too bad. Right although the function looks a little bit of symmetry, but it captures the general behavior of the Original function f of T now one thing I want you to at least observe is that peace sign X by but you can see the denominator. It goes down not so fast. The denominator here 1 3 5 7 and so on so it is kind of going down that a great night all the 1 over n when you increase the number of times it goes down as 1 over n 1 over n is so something really fun. Right thing is fast. You can if you want to make it. Of course, if you add 1 1 you have Onyx Series has an even compared right, you know the harmonic Series this one has to has three. As for and so on it doesn't come. All right, so is It is not something very nice. As far as it doesn't go down fast enough. Okay, so but this is something I want you to remember. You have something not going down too quick and that means if you want to represent your wish to know function, you need to have a large number of terms a large number of terms. So at least you have some observation that we can summarize more times give better approximation. At the point of discontinuity, we have the point of discontinuity. And also there is a help hi-hat. So minus 1 that is on both sides the series converge to one half which is an average value of The left hand and right hand limits at that post now. This is something worth looking at again. You see this is the original value at the point t equal to zero. When you go a little bit to the right, it has been one the average of this Gap is this value? T equals to 0 put in 0 0 0 there is actually one half exactly. It gives you this the value of which is half of the Gap and also that needs to plus Phi cos Pi. We have a zero value. On this side. Yes, it's always give you half of the Gap because you see when you look up a little bit further. On this side. The function will be here. And then the function will be there. This is just halfway halfway halfway of the Gap the Fourier series gives you the value. Which is just the average of the jump. So this is in fact a fact of who is huge when you encounter. Something with a jump in full series always give you the mean value of that job. You can also see what songs that always signed Charles protect the concept. I'll present in the scene and only odd harmonics hopefully on trying to tease out the coefficients the end and virtue of a scientist are proportional to wonderful and I've been should get and hence slow. Very slow Let's look at another example. This example is T to power T defined from - Thai to Thai. So this is an exponential right tree for this function. It is periodic but there are always missing value and high - pie pie. There's no value on so the function in fact has missing values. And any point so here instead actually an empty circle if you want to indicate that that is also an empty circle. The function has an have any defined value at those points? But this doesn't mean that you can't compute the Fourier series and Fourier series that thing is that you're doing too great when you do integration even with some missing points in person at all, right? If you have some missing voice often that's on that. So when we try to compute the area under the curve here or here even without some points you still can obtain the result. This is so true. If you want to do something like Taylor's expansion about the point is that point has missing value and you will be in trouble. You can expand about a point with missing them, right? You may recall that for Taylor's expansion. For Taylor's expansion. Let's say f of x plus h is like A plus h is f of a plus about the point a and then we have H over 1 factorial F Prime of a and run over so so you see If let's say the function is defined at the point A Plus KH you come it's extended about the point a because this has to be fine. If it is a Taylor expansion and you also need diversity. Exist in order to compute this term so it does requires a lot of nice property smoothness of the function at that point of in Spanish come here. It doesn't require those conditions. Okay. Let's get back to the problem of finding the Pella fully series of this. 
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38You're doing it quite mechanically, right? You just integrate. So if you have some calculators that can use integration. That was nuts. Okay, and this is f of t? This is a puppy. Sorry. This is integrating from minus y divided by Handsome time. Now this of course for you, perhaps a requires a few steps to come up with this concept. So I don't write down the answer and I have used a hyperbolic function hyperbolic functions if you heard about this. Give me your calculator, right but you probably have thought you said before that has a bullock sign. So basically it's just 
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40Maybe find something called. 
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42This is my cousin is you can define something called coach. 
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44All right. Let's say these are hyperbolic hyperbolic sine function. I usually that is pronounced as sharp. When I was asked coach Why do people want to do this is because you can recover a lot of nice properties. From trigonometric functions. Yes to the first question is whether we want to use Conquest somebody else. You sound like somebody who can express it as exponential of int but I don't know. Why don't you want to do it here int is equal to what goes int? Has high side and tea. Well, you can do it that way and then recover the real part, but I suppose people want to go into that right because they are very the second question and I think he prayed to get. Google great white shark ozai. Well, the good thing is that if you differentiate a try you have coach Then you can integrate faith is the integration is the reverse of 
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46Yes, yes, you can have something similar but I I forgot about all these things. That's a good thing about the because consign because now I have a cosine squared theta sine square feet and using them. I learned it well done. They formed a while younger than you. How do you expect so you can You can. Convert the triple no metric identity like this. Whenever you see a square you change some then you can recover you can Define hyperbolic tangents have a garbage can only say no. No, I don't want to go into that because I just want to still they finally am but this is I just want to Use this as a more compact way of describing this this time. I use I take the one half year 45 or two and I create this sine function. So the pm10 you can also perform the same similar integration and come up with this. Okay now this time When you integrate using the formula of 4 a.m. You don't need to separate out the a zero because it works for zero as well. Okay in the previous example when you fucking zero the results. Was 25 but here is a and even worked with zero. Okay, so we use we compute A and N from 0 1 2 3 and b n n equal to 1 2 3. Okay. So now this is the answer, okay. So sometimes if you are different only now, let's look at this. Let's look at this now sinks. The coefficient here. I mean if I want to add many terms and see how fast. It goes down that such that I don't need to sum so many terms in order to achieve a level of accuracy. You can see that this term goes down quite quickly. This time is order 1 over N squared right the swallow this term. This is nice it goes out like this. This one looks like it is but in fact there is a 10 in the numerator. So this one also follows this or that and so it means that you need to some many times in order to get to a level of baby every private high accuracy of the representation of the function. So what is written here is about what happens if I if you approve or you give me a function just a function over the range from - typhoon cap. Can I do a Fourier series representation? 
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49And what would happen if I do that? So the question is like is okay. Let's say this is from minus 5. And some of you say that I did. How function like this? Okay, let's say This is f of T. So it is not a periodic solution. But I can still perform this sort of integration and find out. The coefficients the Fourier coefficient of the sun because what I need Is the function from minus High to climb without even knowing that it is a really periodic function, but see what it gives you a function over this range from minus pie that I can stop doing the computation on the right hand side, right? So this made this stuff of the over this range. I can compute all the coefficients but then what is shown on the right hand side is actually when you compute it in corresponds to a periodic function from high to High 2 3 5 so it automatically create. I guess that's you. So you will it will automatically create a function. Like this like So it would create the other portions for you. Because what you see on this side is a periodic function. So on this side it automatically creates. The extension Periodic So this is often called this cut. And also if you accept decide this correspondence that periodic extension of what you provide over from either side. 
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51No. Let's picture. That's - type somewhere here. Hi. Hi. You have the well many times right? The lowest number of times should be the red one. And then with more oxidation the next one would be the dream and then the yellow and Then followed by the largest number of terms for response to these. So that is the approximation behavior that you can observe here. Well, you can try yourself with some function given this is a The example I was this wrap is stays on only three hours using this topics Maxima. You can freely download from the internet and installed in your system computer, which is very useful. So I have computed this is the a no term any not and plus the other X 5n up to five. Alright, so down actually many times you have cosine terms also from 1 to 5 Seinfeld also from 1 to 5 and also the constant term and that protein over the range from minus 3 times 5 to the Past. So this is you can work. You can just change this number. If you like to 10, then you would have the summation up to 10 the higher the number of terms. 
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53Oh, no, it's just that for example you give me a function over this range without telling me Works outside, but you just want a free series. Representation of that function over this way, but the Fourier series would give you the rest because there is a periodic function. 
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55So I encourage you to download this and try for yourself and you can try different functions. Now, this is related to what we have just explained. How about convergence? We have seen quite a number of properties. Why is the number of properties? And of course, we don't have any proof but It's just something that you probably should know when you apply sneeze. First of all more functions can be represented by furnace used and fell asleep. I haven't explained in the beginning somewhere. For a series cousin required continuity of the function. It doesn't require differentiability Taylor series requires. So it is small strange if for function to be expressible as a Taylor series that I fully seems You can see that in the Some people just know the function can have this invoice can have trouble. All right. You can feel the stress of please. Take a serious requires infinite differentiability at a point. If you want to express Taylor function in terms of an into this Taylor series that has infinite differentiability Fourier series can be a price to discontinuous function. Now, this is another point that is quite useful to though approximation. Oh, You know that the Taylor series approximation we have seen tries to use information at a particular point, which use the approximation can be very very accurate when you are looking at a point close to that the point of expansion. But If you go a little bit further away, the approximation can be very bad. But for E-Series as you can see from the previous example. It tries to spread out the earth. Now, of course, you can see that Sarah here and there but it's more or less. Get up. You don't have a thorough concentrated at the certain well doing further away over this region doesn't mean it will give you larger. So it is sometimes it is a better approximation for that reason. So the distribution of era for truncated Fourier series that it's a foolish. You miss with only a finite small number of permanent paint and Avera is more evenly distributed along the domain of approximation. So that's a good thing. Now this is a very interesting example. We have seen that I think when in chapter one I use this example to show that this function is Differentiable everywhere including at the Orange So if you expel the taters. About Peter series about the point 0 you may recall that you always find Ciro first derivative 0 second derivative and so on. So for this function, you probably still remember we have the property that F Prime 0 is 0 f double Prime of 0 is 0 and any SN X derivative so Taylor's extension does not represent the function when you expand it about the points you okay? So Taylor series representation of this function doesn't exist. I'm going to take off that phone. Now, what about Fourier series? If I use for racing I Can approximate that function using? End up to seven Over the range from minus Chi because this function I look at it from - tight high and then it it is a periodic extension of Yes, yes value is 0 yes, the value at that point is approximated very well. But not because the Taylors. Is this if you want to do a penis extension about the point 0 what it means is F0 plus page essay for that function you have F 0 Plus S10 H 1 factorial and so on right so forth Yeah, and so on but the problem is that this is 0 and it is 0 that is 0 and everyone seems so that is seen but this is definitely not zero right? No matter how many terms no matter how many times you can never get back? Yes. That's it. I would expect that. The problematic point is at zero for this social. That is the only point you come to a tailor in Spanish. The other points okay, but this is just for illustration that even if you have a functions with many well with this is infinitely differentiable at the point. You can stop represents the original function using Taylor series that that is the idea. Let's see. This one I can use for S series The Red curve corresponds to the blue one because the blue one is the original function f of T. The blue one. The red one is different things because This curiosity up to this point - Hi and then it repeats itself. So you can still perform very accurate approximation quite accurate. So now we have 4 some. The knowledge is let's say if you have a real valued function of active is said to be piecewise continuous on an interval is inside. The function of that is a close intimate right hevesy find that continuous at all points, but a finite number of accept finite number of points in this is so the function is continuous except I'll final point. That means the function can have this continuous job of The right and left hand limits means that if you approach a point T naught this is from the right side because it's plus you approach the function from the at the point T know from the right hand side. That is what it means. That's the live. This is a location right here. You can link you can also approach music You can have a function that is discontinuous. At T not possibly and you can approach in this direction. So the limit Taking value is f t naught plus if you approach from the left hand side and this value is T your minds. Well, of course if you have a function. Two newest and left hand limit right hand limit will be people. Right and the function is defined at that point. So you have another situation which is something like this. In this case This limit exist That limit also exist. So the two limits exist then the function at this point has a limit. Well the function and avoid of the limit we don't care about the feathers. The function has a value, right? That is an important thing about limit. We only look at what happens when we approach that fault. We don't care about what happens. It's Earlier that for but of course there is a very nice situation. In this case This side exists That's I also exist. And the function f at T naught equals to f t naught plus also equals to f t not minus right this left and right-handed left-handed miss the same and also equal to the function value at that point. That is what we call. Continuous at this point. This point is continuous. So That's the nice part continuous. At Tino It's the this case. Yeah, we only have limit exists. 
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57Yeah, the West case. Limit does not exist. 
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59But you still can have left hand and right hand. Okay the sea level. Okay, so these are just the terminology. We need to start off going this fucking song. 
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61Okay. So even by looking given a function like this you can graph it. And now with that description you will know when cell dysfunction is we are only looking from minus 2 to write the function is only defined over this region outside the okay so over this region you can see S 2 minus S 2 minus means here from this app - on the side so the limiting value is 4 It's limiting value is 4 that is no. Definition of that right hand limit as to because the function is not defined outside is passed. So there is no meaning of approaching 2 from the right hand side. So in this case f so this F 2 plus That's not make sense. Right? Because the function is only defined up to q1 minus means on this side. So the value will be too. Wildcats from this side. The function value is 1 so you can work out the rest and so on so here the function has a missing value at 0. Not confined at the points here, but even it is not defined at 0 you can still have have a right hand limit at hand left hand limit exist, right? 
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64So function value is not required to be defined, but you can still find it. Limit left hand and right hand 
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66So you can also Define left-hand and right-hand diversity. So you are looking at the slope from the left hand side or from the right hand side. So looking at the previous picture you can see well if you move the say at this point if I move towards this direction, and I look at the slope. The slope of the tangent So I move in this direction and you have the slope here the slope here and then you can essentially in the end you can draw attention. You can obtain a tangent. 
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68And that is the Right hand. Write the derivative in the right hand for the slope from the right hand Direction. So at this point you can also approach it this election the right hand and the left hand derivative of course will be sealed. So any point, you can also Define left hand or Right hand red But the important things is that when he finding the safe, let's say this is the right hand derivative. This is Ft. Hostage H is a positive number so that it approaches 0 from the right hand side and subtracting the limiting value. This is the functional value right hand limit values. If you look at it from the other direction, this is - all right, because we approach from the minus Direction. It just still positive here actually, sometimes you may not like to have this version you probably if you want to you can convert everything with a minus sign in that case well for the second page because you are not perhaps Not used to - picture - Stitch there, but you can change it. So let's change it to let's say Delta with a minus. Okay, so what Delta equals to minus H so you can have this as a belt? And Delta is negative. And then this is pastels and here H tends to 0. Actually that means from the positive side because it is always positive but You have Fanta from the negative side Delta is negative Delta from the negative side. Sometimes you can put - right you can also do that is approaching from the left hand side. That is that is Also possible Talked always doesn't know what is written down originally is correct. So finally when you have the left hand derivative exists right hand with the rivet images and also equal state. F Prime T. Not then well when ever this F Prime X is means that they are. All right. So you have the left hand right hand limit exists and the same and you have exactly what we mean by the existence of the derivative at that point. So Example just now we have a few cases. Basically, what it means is at this point, if you look at the slope, if you are talking about the right hand derivative at the point one, look at this rope left hand derivative at this point. You look at the slope here at this point. You only have one side. This is the right hand the left hand derivative. So you look at the scope here the limit of the slope there and so on so it is a very intuitive. So the limit of the slope is different approaching a different direction gives you the last the corresponding left-hand or right-hand lips. So the convergence theorem, let's Talk about this. You watch baby is too much for me. That's why I still have two or three or four hours more. So saying yeah you if you think you have all the time. Look into this actually is my first of all the good thing is that it will not be examined. You need to know some facts the fact is that of course this is described in a very mathematical way, but the thing is very very simple. No proof. This is a whole it's a convergence you remember that we have sometimes will function the Fourier series involves example, you can see will converge to the real bad. Let's hope this one is here. It's just if you have a What is sex over here? Everything is just about if your function has a gap. Fully series will keep the cat about that's all no matter whether you have a value here or there doesn't matter. So let's say if you have the function looks like this Fourier series will converge to the mean value at that point. What does that mean? If you have two three. Representation you substitute the value. Well, the KT value here, you will come up with this bag half of that right now. You may say what happened if I have A function which looks like this Then what value for a series with give you at this point is gorgeous Point. She know right? That's too long. So the The theorem actually says it doesn't matter whether you have animals here or no value. Fourier series gives you half of that boys. So if you have something like that with a missing value for a series will fill the pulling for you. What that's what Fourier series of if you have an empty but the perp is having left hand right hand limit the same button missing point three cepheus give you the mean value of the mean value is exactly the same value on both sides. So if you have something, of course the nice thing the nice situation the function is continuous at this point. For a series Will give you the same way. Of course. So your syrup. Yeah disciple he say. 
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70Yeah, I think so.