Fourier Series examples and solutions for Even and Odd Function

Fourier Series examples and solutions for Even and Odd Function


Hello students, today I am here with a new thought and a new topic today’s thought is that light lamps for others and your path will also shine Sprinkle positive thoughts around you and that positive energy will help you shine bright Today we are going to discuss Fourier series which is important for engineering students and BSC students and other government and competitive exams first-grade school lecturer or second-grade teacher or any else or for higher education like NET, JEE, etc. today we are going to discuss what is the Fourier series? and and how we solve them we will solve some numericals based on that so what is Fourier series so, students, we have some new words here what is a periodic function? it is that function which repeats itself after a particular interval for example these all are periodic functions if we see the series of sin x it is of this type so as you can see after a particular interval the series repeats itself these are the sin waves it is repeating itself similarly, if we see the graph of cos and tan after a particular interval function repeats let me tell you we study in engineering or higher mathematics we can represent series algebraically by Taylor’s theorem and Maclaurin’s theorem the series of sin x is here we have represented sin algebraically in the same way, if we have a function that is algebraic, like all these functions can be represented in series in terms of trigonometric functions like x can be represented in terms of trigonometric functions and that is the concept of fourier series here we will learn, how can we represent algebraic functions in sin and cosine terms now you will have a question that sir algebraic functions are not periodic then how can we represent it we can represent the algebraic function too in periodic functions here we will have to define that it is repeating after a the particular interval of time so we will discuss the concept in questions so students in order to calculate Fourier series the formula we have is this is the formula we have for Fourier series if you have to calculate the Fourier series between -l to l then we will use this formula here the value of a0 is formula for an is and the value for bn is so students we have the following values we will solve an, a0 and substitute there and then we get the Fourier series and the interval if -l to l if you are asked to find the Fourier series between -2 to 2 between -pi to pi whatever the interval accordingly you have to calculate for example if you are asked what we will do is wherever there is l we will keep ‘pi’ keeping ‘pi’ the Fourier series we will get is this is our Fourier series between -pi to pi we kept ‘pi’ in place of l the value of a0 will be value of an will be value of bn will be s students what we have done is wherever there was l we substituted pi this is the Fourier series we got now we have a concept of odd and even here which you should know if the function is odd then what will happen and if it is even what will happen let me clear you the concept of class 12 if it is an even function then what happens and if it is odd then what happens let me tell you the concept of even and odd as per class 12 the concept of integration in any function if we keep -x in place of x and if it is equal then it is an even function for example: cos x is even we denote even function with ‘+’ sign similarly in any function if we keep -x in place of x and there is a negative sign then it is an odd function, For example: sin x sin x is an odd function x is an odd function these all are odd function you will say how this is an odd function so let me tell you if we keep -x here it is not equal it is equal when the minus sign comes out so this is not equal if it is equal then one is a positive one is negative this is an odd function, for example to check xcosx is even or odd remember the concept of plus and mminus we denote even with + sign and odd with – sign here x is odd and cos x is even function so plus*minus=minus so this is odd function same if we take xsinx x is odd and sinx is odd but minus*minus=plus so this is an even function now we studied in 12 class if any function is even from -l to l then in that case, its value will be equal to so this is the concept of even function but if we have odd function from -l to l is an odd function then its integration is always 0 in 12 class these are the rules of integration we studied now in Fourier series if the function f(x) is even if the f(x) is even or if the f(x) is odd if any function is even then what will be our Fourier series and if the function is odd then what will be our Fourier series I am telling you this between -pi to pi so students, if the f(x) is even we will denote it with a plus which means it is an even function and if it is even what is the property? then what you will get is the value of a0 will be students, let us talk about an now this is our even function and cos x is an even function so an will be similarly we will get bn as this is our even function but we know sin is odd so + * -=- so value of bn will be 0 if any function is even we will calculate a0 and an the value of bn will be 0 remember this concept if you will solve it, it will come 0 but will waste time substituting the value we will get the Fourier series as now we will talk about odd if the function is odd then this f(x) will be – and if this is odd its integration will be 0 so students if the function is odd then our a0 will be 0 this f(x) is odd so minus * plus=minus so our an will also be 0 talking about this function here it is minus, so minus*minus is plus so our Bn will be as this function is even So our Fourier series will be so students what i told you is this is the formula for our Fourier series between -l to l this is the way we calculate an and bn and we get our answer in series form if the series is between -pi to pi wherever there is l we will keep pi then I told you what is even and odd function we discussed the difference we denote even function with a plus sign and odd function is denoted by a minus sign if a function is even then there will be a0 and an but bn will be 0 always in the odd function a0 and an is 0 and only we have the value for bn so, students, we will discuss two questions one on even and one on odd so here is the question first of all we have to see the given function is even or odd as you can see that this function is odd and why? if we keep -x in place of x then we will get the negative sign you can see and -x this function is odd and if it is odd then in that case our series will be you have to write it as shown we have discussed in the formula in odd function a0 and an is 0 and the formula for bn is now we will keep x in place of f(x) now we will do its integration here so, students, I want to tell you as you have studied in 12 class how we do integration by parts let me tell you again if you are given two functions solve as shown this formula we will use here you should know this which function to take u and which one to take v we have a lecture on that, I will upload when needed just now we will use ILATE here where we get to know which one to take as 1 and which one 2 I have shown you the formula here the first function will be x and second function will be sin(nx) solve as shown solve as shown we will do its integration solve as shown here we have one more concept that you should know the value sin (n(pi)) is 0 and for cos is this you should know because you will have a question how it is 0, here n is a natural number if we keep n=0 sin 0=0, if we keep 1 sin pi=0 it is 0 for all these values whatever the value of n we will get 0 if I talk about cos here n=0 value=1 n=1 value=-1 and n=2 value=1 we have alternate + – so we have this value here we will use it here where it is x we will keep pi this is our first limit now keeping x as 0 solve as follows so students we will get the value as the value of bn will be value of cos (n(pi)) substituting the values and solving this is the value of bn we got we will keep this value of bn here then our series will be we will keep the values we have got the answer but you can solve further in the form of series keeping n=1 then we will get the series as keeping n=2 this way we will get our series so this is the Fourier series of f(x)=x between -pi to pi we have the next question here If in exams you are asked till Fourier series then you will do it till here but if you are asked to reduce these relations you have to solve it further first of all it is an even function because if we keep -x then there is no change so this function is even then the value of bn will be 0 so our Fourier series will be in the form we will calculate the value of a0 keeping the value in the place of f(x) we will do the integration simplifying it further so this is our a0 calculating the value of an we have the formula as solve as shown we will do its differentiation solve as shown simplifying it further using ILATE so, students, you can see keep x=pi value of sin (n(pi)) is solve as shown, taking the next limit getting the values as follows you will get the value as this we have got the an here we will keep the values of a0 and an in the function we will get the f(x) as value of an will be let us expand it a little this way you will get the values so if in exam, we are asked to find the Fourier series then we will solve it till here but if you are asked the relations too then to prove this then you have to solve it further let us move forward value of f(x) is if you have to prove you will keep the value of f(x) value of f(x) is write the expression as shown this is our Fourier series put x=pi then value of cos pi is -1 you will have a confusion here that how I have written x=pi here so let me tell you if it is cos then keep either 0 or pi because it is not 0 and if it is sin then keep pi/2 next we will simplify it simplifying it so here we have got our first relation similarly we will do x=0 because we have to prove 3 relations this is our first relation keeping x=0 then we will get the value as after simplifying it we will get keeping values directly if you have any doubt you can drop a comment i will explain it to you we will add these two relations now you will say how we will know that we have to add. we have got 1 and 2 relation, in 3 relation we will add so it will cancel so we will get because in 3 relation we want we will get it by adding them we will get the value as so this is the relation we will get so students, today we discussed Fourier series we defined Fourier series and what is odd and even functions and we discussed two questions here one for the odd function and the next question was of even function I am getting positive response from you all and I am feeling motivated your responses motivate me to bring more quality content please subscribe to my channel and do share my videos

100 Replies to “Fourier Series examples and solutions for Even and Odd Function”

  1. Thank u sir 😊 it's help more .
    Make more videos like this.
    Good explanation, it's understandable .
    Again thanks alot.

  2. What an astonishing moment to get this smooth and crystal quality education for free… Would love to share it … your majesty… Love it !!! Thanks, Teacher!!

  3. Its difficult to see the writing clearly, sir pls improve the camera quality and try to write more lagely so that its clearly visible

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  5. Sir your formula of F(x)in fourier series is incorrect ao÷2 or a(n) ka formula bhi shi nhi hai but you teach well sir reply

  6. Sir There is a problem that in even function the formula of Ao is 2/π and you are writing that formula with 1/π in the interval of -πto+π

    ..
    But sir the series according to book is ao/2 and you are writing only ao
    And again the value of ao acc. To book is 2/3 πsqr and your value πsqr/3
    So i am confused so tell me the right way.

  7. !!.. Special Thanks Sir ..!!
    Today My Engineering First Semester paper
    And Sir Mene Sbhi Topic Aapke Video Lecture Se Hi Complet Kiye The Or Paper Me Esa Koi Bhi Questions Nhi tha Jo me nhi kar paya Mene math m acha paper first Time Kiya Hai Meri Life engineering student hone k bad bhi MERI math weak thi jb se apke lecture Dekha Hu ab me kisi or ko bhi smja skta hu itni Ability aa gyi hai mujme

    !! Thank You So Much sir !!

    Bhavnesh Verma
    Government Engineering College, Bikaner
    Rajasthan (334001)

  8. Thank You Sir for your lecture on Fourier Series examples and solutions for Even and Odd Function !

    It was really Helpful !

  9. you teach good…but I have a question ,that in every interval of even or odd function an or bn etc become zero? or just in -pi to pi interval

  10. I easily understand the topic FOURIER SERIES after watching this video. I'm a student of bsc maths hons. CU 3rd year. Please make a video about TENSOR ANALYSIS.

  11. Thank you so much sir… By watching this video based on Fourier series I don't have any more doubt about this chapter.. Thanku

  12. Sir xsinx even fxn hai to isma bn नहीं निकले ga पर BSc 6th real and complex me te bn निकाल rakha hai sir please batana

  13. Sir, Thank you so much for ur constant support for us sir
    It's very useful to me.
    I can understand the steps easily sir.
    TQ sir

  14. Sir agar ap thoda speed slow kar sake to bohot acha hoga …..it would be very easy for us …..and thanks for the amazing content ….🤙🤙🤙

  15. Your videos are helping sir as we are studying from home nowdays… Thank you sir for conveying your knowledge…♥️

  16. thank you sir for this really appreciate your teaching skills. Good luck ahead sir make more and more videos. again Thank you sir

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