Parseval’s Theorem

parsevals theorem
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okay as we all know thanks to Foley we can read function f of X as a trick series on Interpol phone negative PI to PI and of course well just call this right here to be the Fourier series and now in this video let me show you guys what pastor felt did here we go we will be looking at the integral from negative PI to PI and we look at the function we square that so I will put our parentheses with f of X inside and then we square that right here th you guys I come to like this result because we can do a lot of very nice things based on this result so be sure you watch each other yet here we go first of all of course not to assume that this right here does converge to the function on this interval and when we have that we can just put this right here for the function so we have the integral from negative PI to PI and then we have our parentheses f of X is just all that so we will have a 0 plus the sum as n goes from 1 to infinity all this which is a cosine + X + b + sine of an X like this and then top again we square that right here and of course we still have to have the TX right here so thats pretty much that here now we will be looking at this as a binomial because we have the first + the second thing right here square that of course we will have the first thing square plus 2 times the first in a second and then plus the second thing squared right everybody in the integral so well pray them thank – sweetie finger goes first one we will have the integral from negative PI to PI this thing squared which is just a 0 square that closed up right so thats the first integral to look at and yes to figure what a serious we have to use the integral for that but dont worry because this right here will be just a number for our situation right and if you want to see how to figure out a 0 a and mbn be sure you guys watch my other video for that by anyway continue next part we do two times this and that and thats in the integral so we add the the group of negative pi 2 pi and then 2 times this and that which is a 0 and then the summation n goes from 1 to infinity and all this which is a cosine and X plus B sine of X like this and then close the integral so its its like this okay this hopefully its not that bad now the third one we will have to add this thing square in the integral which is the integral from negative PI to PI and heres the deal we have to do this square and this is a summation and khrysta to this one carefully but dont forget whenever we have something square we can just put on itself times itself right so I will write this down right here first which is the sum as n goes from 1 to infinity and then we have a cosine and X plus B sine of X like this in the first summation and then I will put on not 1 the summation in blue but because Im putting down another summation be sure we use a different index here I have an let me use em right here and dont let the end what you bother you they are just determined variables and whenever you are multiplying two series or two summations be sure you have different index right so this part we will just be writing

this down right here by change audience two Ms so we will have a M cosine M X and then we will have the BM sine of M X like this cause that and dont forget this guy right here is still in that integral so we still have to have the DX all the way at yet whoa okay cool now lets figure this out first one easy why because yes I know Israel is in terms of integral pie its just a number in our situation a zero square is also a number right when we integrate a number from negative PI to PI of course we just get two pi times this number so the first integrals very nice is just 2 pi times a 0 squared of course you can do this well right now for the next one what well do is because we assumed that this ray here can purge we are going to interchange the integral and also the summation and because we have the two a zero its just a constant multiple let me put that down right here first we add to a 0 and then we put down the summation first which is and goes from 1 to infinity and then right here lets do care the integral from negative PI to PI so Ill put this down now which is negative PI to PI and then of course we have to write this down here which is a cosine and X plus B and sign of an X like this and then of course thats half that DX right here and before we look at this one lets finish this this is actually not that bad at all okay I would write down some notes right here for you guys again a an ambien in our situations they are just numbers especially when youre on the x world dont worry about them I just like a custom multiples focus on the integral from negative PI to PI cosine X so let me put this down right here for you guys note when we have the integral from negative PI to PI of cosine or lets just do X first when you do this you can do this in your head real quick you will get positive sine X for the integral right and then youre putting pipe hey hey thats your row you put in negative PI hey thats also 0 so all this will be 0 but guess what if you have cosine of any constant multiple times X well and has to be a whole number well and we are having a whole number I want 2 3 4 5 up to infinity oh thats aside no oh well n has to be a whole number anyway cosine X integrate that you will get 1 over n times sine of annex pudding pie sign off and pie when n is an integer is zero putting negative pi we also get zero so everybody is zero nice now lets look at the other one negative PI to PI we have sine of an X this also is just a rope why because sine its an odd function and when you have an X in the side thats to an odd function when you integrate a function for negative PI to PI you get zero so in fact you have two in Stuart times summation of bunch of zeros everybody here the second term to zero so bye-bye for the second term good now for this part here is the deal as we can see we have a summation x another salvation and let me just give you guys an attitude real quick remember the times when we two integrals for example you have integral from 1 to infinity of some function at the f of X DX right this is a empowering you okay

times another one as a integral from 1 to infinity but this one has a G of Y dy which is another integral integral times the integral in this case what we can do is we can put the integrals together integral from 1 to infinity and then integral from 1 to infinity and when we put them together be sure we just multiply the insides together so we have f of X times G of Y and depending on what you want to do but in this case it seems Im cutting down the dy first and then the Dax element outside right so this is doable thats just the idea for this part so what Im gonna do is I look at this Ill put down plus and yes I will be switching the integration and also the summation but not yet I will actually put this and that well Ill put down this first all the way in the front so the sum as n goes from 1 to infinity and then I will put down the sum as M goes from 1 to infinity and we can do that just right really similar to the idea you can just do this for the topo integrals and then what well do is well just multiply the insides together so thats the apartments right so here we go this times that we have a + cosine NX times am and cosine MX and they will just continue and see this is like the cool boil this is very nice this times that we have plus a n cosine a next time x BM sine of MX and of course we continue and then this times that by here we add be a sign of an x times a a.m. cosine MX and lastly of course we do this times that we add pn times sine of an X and this is B M sine of MX like this right so we can actually do that with the double summation now right here and then remember we have this integral the weight in the front isnt it so be sure you still have the integral right here so negative go from negative PI to PI and of course thats also called the DX I mean read what well do is of course put inside integrating from negative PI to PI and then right here I have 2 DX here and then I will close that for you guys so I did put on a DX but you guys might not be able to see so let me just okay so this is hows that No okay so I did have the DX and that will have the parentheses just in case if somebody cannot see you I just want you guys to make sure you have the DX and that all right cool now this is rather bizarre because we have one two three four four things in this integral pie its actually not that bad at all again a nama and all the coefficients just dont worry about that pay attention to the sine cosine functions and I will write on some notes right here for you guys first thing first write note lets look at integral from negative PI to PI lets look at this one first cosine of M x times cosine of sorry cosine X times cosine of M X and if you guys have seen my video on getting the formulas for a zero a and PN yeah its pretty much that part right here again let me tell you guys this right here when we have this integral we only have two possible outcomes its either we get zero or we get PI this will be 0 if n is not equal to M and this will be PI if and is equal to M so this is very

nice because when we have the summation in blue when M goes from one to infinity we actually just have to consider one term namely when M is equal to n pretty here so we actually just have to deal with one summation you will see because the other ones will be zero anyway thanks to this right here and then I will pull down the other one for you guys as we also the other one is when we have integral from negative PI to PI as you can see here we have cosine times sine right so Ill put this down here for you guys so when we have cosine of n x times sine of MECs DX guess what this right here is always going to be zero because cosines even science art even times art is art so oh no here we have an odd function integrating this over negative PI to PI we get zero very nice so this is true for all and and M very nice and of course this situation sine cosine its right here as well dont let the n-word Ampata you sytem need to write it down yet the last iteration is when we have sine and sines right so integrating from negative PI to PI we have sang off and x times sine of M X DX and again for this integral here we have two possible outcomes its pretty much the same as this one so put down 0 or pi this right here is 0 if n is not equal to M this is PI if and is equal to M so thats the idea so real quick based on this right here especially the middle one we know this and that will be just 0 when we integrate from negative PI to PI so thats nice well the only thing that we have to consider the blue part is when n is equal to n because thats the only part we will actually have some values for this integral right so let me just put this down right here for you guys all we want is which is put this down here all we need is when n is equal to n like that right you see when M is equal to add so here is the idea M goes from 1 to infinity so that means 1 2 3 4 5 so and at some point you are going to hit and put the end here all right the other case you dont have to worry because when you integrate the other case to be 0 anyway so I just have to consider that part let me write that down right here for you guys which is a and in blue and then cosine of X just that part and then dont forget we still have some Asian black which we have to multiply this power hdan and also the cosine and X in black so I put that down and cosine an X right here and again dont forget we have the summation when n goes from 1 to infinity and dont forget we have to look at the integral as well so consider this part and then the integral from negative PI to PI DX and now heres a beauty if you look at the red part which is the cosine NX and also the culture and X again MN and are the same right here the red part right here thanks to this we have just pi so this is what were going to get pi is just a constant multiple so I will just put that all the way in the front of the summation so we have pi and we have this sum as n goes from 1 to infinity and then dont forget we have a n times a and inside which is just a

the wonderful so just put this down right here a and square cool now we do the same for this one right for this integral does einsteins situation and when we have that dont forget we still have this summation + the next summation or just reading stuff like yo again this will be N equals 1 to infinity its a same idea when M is equal to Nu puddi and into the MS right here and then you have all this Park which is B and Stein and X and then sign off an X also the pH but this in blue so this is what we have right here and dont forget integral right everybodys favorite negative infinity to positive infinity I mean negative PI to plus D PI and then we have the DX here and you know the do this and that the red power give us PI so thats put on the pie first all right so this is equal to PI and then we have the sum as n goes from 1 to infinity BM BM which is just BN square very nice Wow so the third pot fine how do you wish its right here this right here its nicely equal to this class the Hat of course we can put the seamstick getting seven just once tamesha right so hopefully thats clear and now let me tell you guys how passive held it right as you can see when we integrate f of X to the second power going from negative PI to PI we have this its equal to this thats put zero okay so just this Plus these right here right and notice everybody has PI so well just divide pie for everybody so let me write down the statement for you guys let me just write this down for you guys here okay let me put this down integral from negative PI to PI again we will be you know putting the f of X in the parenthesis squared ah yes right well this is equal to this part but I want to put a PI by T by doing the PI on both sides so I will have that one of a pipe right here thats just a tradition people do that – Chris oh I didnt do it other people do just follow so I will just put down two times a0 squared like this and then of course this pie right here already okay so we just have these two things I will put it in one summation so we add the sum as n goes from 1 to infinity parentheses right a and square and then B N squared and that is very very nice let me just write this down nicely for you guys like this this is the car set house 0 very next and this is what I use when I take f of X to be X square and I figure out the coefficients for a 0 a and MBA and I put it here and I get read in a sum the sum as n goes from 1 to infinity of 1 over and to the first power you can also do this with then to figure out the sum of 1 over N squared 1 over and to the 6th power know that that will leave that to you guys right well question for you guys is it possible for us to use this right here and also the Fourier series to figure out the sum as goes from 1 to infinity of 1 over cube is this possible by based on the things that I have demonstrated for you guys well leave a comment down below let me know by anyway hopefully get so like this video if you do please give me a like thank you guys so much and as always thats it

tags:
Parsevals Theorem, Fourier Series, trigonometric series, blackpenredpen, math for fun, hard calculus problem, Parsevals proof, Parsevals identity
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