EXAMPLE 1 OF FOURIER SERIES EVEN (BY MR ONYANGO)

so so in our example one we are now given the periodic function f of t is equal to pi minus t when t is in between negative pi and zero so we have to sketch the graph for three periods and when you are told through period you sketch the one period then the other periods you can get them by having a repetition of what is going on so but to determine the fourier series part 3 from the fourier series by letting t to be equal to 0 you show that pi squared over 8 is equal to summation and starting from 1 to infinity 1 all over 2 n minus 1 squared squared so you do that letting and then you get your result so let's start in our solution by sketching the graph for three periods so when you have a function which only carries up to t power one then your function is made up by portions of straight lines if you are ready spi minus t or pi plus t or t only provided you are not going to t squared or t cubed you are dealing with straight portions of straight line straight line but when you have t squared in your expression then you must know you are dealing with a curve a curve but here you are just dealing with words dealing with the straight lines so in a straight line what do you need you only need the end points of your straight line then with your ruler you'll make up the straight line line so we start end points of our straight line this is first portion when t is minus pi here when you insert minus pi here it will be pi minus negative pi so f of t will now be 2 pi 2 pi so we say now when t is equal to negative pi f of t will be equal to 2 pi when t is equal to zero we are now taking this end we started from here now we are going to another end zero when t is zero f of t will be equal to pi so when t again is equal to zero when t is equal to zero input zero here f of t will be equal to r pi and when t is equal to pi our f of t f of t will be equal to 2 pi 2 pi so these are the endpoints that we need to sketch and after sketching we input them after sketching then we are going to use our graph for other purposes so we have something like this [Music] here can be pi negative 2 pi negative 2 pi negative three pi then we have here pi we have two pi we have three pi [Music] we can have here pi and we can have that 2 pi this is the graph of f of t and that is our t axis now this is point negative pi 2 pi so when we have negative pi we are getting we are going up to positive 2 pi we are there that is the point when t is zero f of t is pi so point zero pi which is this point that is point zero pi five when t is zero f of t square so point zero pi which is the same point and when t is pi f of t is two pi so we are going upwards we are there so that is it so we join those points with the straight line straight line straight line and after joining those points with straight line that is only for one period one period now you are told you are supposed to do it for two periods and then you say ekiti nancy i killed you so in my hands i killed you you made a now from this we have another period we can now insert here is now [Music] that [Music] so that is our function for three period periods now that graph we are also going to use it now when we reach part 3 of this question because if you don't have your graph in place there will be no way you will answer part three of this quest question now from there you can now answer part two now when we check our function because our function started from here up to there these are just other periods when we check from here up to there is that function even or is it old that function is what it is symmetrical about the vertical axis so it is an even fun function so after knowing it is even we say in our part two the graph is symmetrical the graph is symmetrical asymmetrical about the vertical axis about the vertical axis hence about the vertical axis has f of t is an even function an even function therefore our bn will be equal to zero this implies our f of t how we'll write its fourier x fourier series representation will be a naught all over two plus summation and starting from one to infinity and starting from 1 to infinity a n calls and pi t all over word l but 2 l of hours the full period was 2 pi so our l value is equals toward pi so finally when we'll be writing our fourier series this will be a naught all over 2 plus summation and starting from 1 to infinity a n cos and t because if l is pi this pi will cancel that at the pi so we'll remain with the cos word cos nt now let's start by getting the fourier series coefficients let's start by getting determining the value of n of a naught a naught now will be equal because this is a special function our a naught will now be 2 all over n we do it from 0 to n then f of t cos n pi t all over n the word d t which will be equal to 2 all over pi we are doing it from 0 to pi so when you want to check your function of t between 0 and pi what is the name of our function of t it is what pi plus t so that's what you write pi plus t then multiplying equals what and t because l is pi then we have here dt don't worry to what happens on the other interval because this is a special fact function whatever we are doing there why we are saying it is two times it it contains the other pro portion so you simply do two all over pi zero to pi pi plus t cos nt dt will give us the full razer result if you are doing it from a general perspective we would have used 1 over l then we say it would have been from minus l to l then we get what we get the answer but by doing this we are just trying now to shorten our working because this is a special fact function without evaluation this is pi plus t this is cos nt if you differentiate this you don't get pi plus t or you don't even get t if you differentiate pi velocity you don't get one you don't get cos nt do you get that so those are two different functions which are not related to each other if they are not related to each other then we can only clear by integration by part by parts it's a question that is a n that is a n not a naught [Music] go back to our function that's why i did not erase this so that you can be able to see between 0 and pi what is the name of our function that one is that okay yeah that's why we are not using this and i explained why we are not using this this one is now consumed into this other one yeah this is a and not a naught so we use our integration by part we let u to be equal to what let u to be equal to pi plus t and we let our dv to be equal to cos and t from your integration by parts formula if you can remember we know the integral of u d v is equal to uv minus the integral of vdu so that's what we're going to use so 2l over pi is here then we have a bracket our du will be what let's write it before we insert it d will be equal to the ta and our v will be equal to 1 over 1 over n sine nt meaning you integrate this d u you differentiate this when you differentiate this with respect to t you get one d u d t is one so the u is equal to d what d t so u v u v we have one all over n n two pi plus what t into sine and t then we have limits from zero to power to pi minus the integral from 0 to pi minus the integral from 0 to pi v d u so it is 1 over n sine nt then we have the dt we close the bracket now when we insert this value here whether is t we input pi it will be sine n pi n pi sine n pi is normally equal to z zero when we insert zero here it will be sine zero and sine zero is also z zero so the whole thing comfortably approaches zero so we are now integrating when you integrate sine energy the integral of a sine is a negative cause a negative will see this negative and it will multiply we will get now a positive r answer that this constant will divide by that particular coefficient of t so this one will now go to n square squared so it will be 1 all over n squared cos what and t cos nt and then we put input our limits from 0 to pi which gives us what 12 pi this one over n squared we can take it out it will not change because we are only inserting the values of what when you put here pie it will because what this will because n pie where there is t will put pie will get cos n pi pi minus when inserting lower limit when you put t here you will put zero where there is t it will become zero which is what one but we agreed we agreed that cos n pi can be written as negative one power n minus one now now when you check that expression we've got an our value over a n but when you check that expression something is going to happen here when n is 1 for example this one will continue being negative 1 1 when it is 3 it will continue bring negative 1 when it is 5 it will continue being negative 1. so for all the numbers this one will be minus one minus one which is minus what two so so it will be minus two so we say this one when n is old when n is old a n will be equal to what a n will now be minus 2 times 2 minus 4 all over pi and squared all over pi n squared now when n is even when n is 7 on na7 so when n a seven what will happen this should be negative one power and every number will be positive one one negative one power two one negative one power 4 1 negative 1 power 6 also 1 1 minus 1 0 so our a n will be equal to 0 that means our a n will only be defined for older values then we go for a naught because we have not established the value of a naught and naught now will be 2 all over l we integrate from 0 to l f of t d 1 d t which is 2 all over pi we're integrating from 0 to pi our f of t between 0 and pi is pi plus what t then we have this one as d dt which will be two l over pi we have pi t when you integrate pi pi is a constant and you get pi t when you integrate t equal to what t squared l over 2 then we have zero to pi to pi 2 over pi let's input this so that we get that answer so this will be pi squared this is 2 over pi this will be pi squared plus pi squared l over 2 which will be 3 pi squared all over what the lower limit will bring all the zeros so the answer here now will be this will be equal to now that 5 will consider that pi 2 will constant 2 so the answer is 3 power 3 pi so document that so we can proceed so our function of t is equal to a naught all over 2 plus summation and starting from 1 to infinity then we have a n cos nt that is according to our simplification because this is a function of period 2 part 2 pi now our a naught we pick it and we divide by 2. so it will be 3 pi all over what 2. now we say summation and starting from 1 to infinity our a n value was only defined for odd values and it was minus 4 all over pi n squared but n must now be up off now we don't want to talk we just want to insert something that when we input our values we'll just bring out all the values so where there is n now we insert 2n minus 1 1 to represent the odd values now this is cause because the n also must now be what an odd value 2 n minus 1 multiplying word t you replace where there is n you put 2 n minus 1 1 this one we only do when you are dealing with order values now we can simplify this and say this is 3 pi all over 2 minus 4 over pi is a constant and can be allowed to pass our operator where our operator is the summation sign so this will be minus 4 all over pi summation n starting from 1 to infinity then we have 1 all over oh this one was squared our n was squared yeah so 1 over 2 n minus 1 which is now squared squared now this one multiplying cos 2 n minus 1 multiplying t that is now the fourier series represented representation fourier series representation now we can write some few terms of for the the series so that we can be able to see it and even if you reach here that is now the correct fourier c series this is just a way you've written it in short but you can also write it in for in full so sometimes even if you leave it there you get your full ma full of max so f of t f of t will now be equal to three pi all over two then we say minus four all over pi then we open a bracket would like to have a summation when n is 1 this one will be 1 all over 1 squared is it also because 2 times 1 is 2 2 minus 1 1 so we have a scenario 1 all over 1 squared cos 1 t because n is now 1 so 2 times 1 minus 1 you get 1 t when n is 2 this one will turn into 3 and so it will be 1 all over 3 squared cos 3t when n is 3 it will be 1 over 5 squared cos 5t now plus others we stop so when we simplify it is 3 pi all over 2 minus 4 all over pi into cos t plus 1 all over 9 plus 1 over 25 cos 5 t plus others that is now our function of t so we are through with part two of our question now we go to part three you are told by letting by letting t to be equal to 0 show that something is happening so we go back to our graph from our graph we are saying f of 0 you insert 0 here and because what you are showing has got still summation sign you go back to expression of function of t with the summation sign in it and because you go back there so this is our f of t remember so whether it's still being started 0 is equal to what 3 pi all over 2 minus 4 all over pi summation and starting from 1 to infinity then we have 1 over 2 n minus 1 which is square squared which is square that's what we are having because cos 0 will be what because 0 is equal to 1 1 when t is 0 this will become 0 which is what 1 now we organize this formula so that we leave this one on that side so we are organizing what is our f of 0 from graph what is the value of f of t when t is equal to 0 read it from our graph the value of t with the value of f of t when t is equal to 0. it is power pi so f of zero is pi is equal to three pi all over two minus four all over pi summation and starting from one to infinity 1 all over 2 n minus 1 square square now we take this one bring this way so it will be pi minus three pi over what two is equal to minus four all over pi summation and starting from one to infinity one all over 2 n minus 1 which is square squared this one will give us negative pi over what over 2 is equal to minus 4 all over pi summation and starting from 1 to infinity 1 all over 2 n minus 1 which is square squared now we move this one away so we are going to multiply both sides by pi all over negative 4 so that when this one is cleared from here so that we have this summation sign have a coefficient of 1 so this will be equal to minus pi over 2 times pi all over negative word for this is what will be equal to summation and starting from 1 to infinity 1 all over 2 n minus 1 which is square squared this one gives us now pi squared all over 8. [Music] and starting from 1 to infinity 1 over 2 n minus 1 which is square square and then we say hence show so we are through with that question you can't document

Channel: JEMSHAH E-LEARNING

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Length: 32min 14sec (1934 seconds)

Published: Fri Nov 27 2020