Session 4 : Convergence of Fourier series.(see pinned comment)

[@DrMathaholic]

At 13:25 , at x= pi, Right hand limit will 0 mistakenly I write -2pi.

[@Tucxy]

Awesome video! After watching this video everything just clicked. Great job showing the graphs the whole time, and explaining practices and theorems having to do specifically with fourier series. I actually replicated this to find S(-22.53) where S(x) is the FSS of 3-x for a 2L periodic extension with L=3. Really glad you mentioned that if the series is continuous at some point then its sum is just value of f(x). Subscribing.

[@DrMathaholic]

Thank you for your comment and kind words.. Happy learning:)

[@sounakbhattacharya9646]

Thank you sir 🙏 🙏

[@DrMathaholic]

Most welcome

[@sudhakarmathsacademy]

Nice explanation sir Thank you.

[@DrMathaholic]

Welcome :)

[@chitrangbhoir7883]

Sir, at 13:32, the value of F.S. at 'x=pi' should be '+pi/2' right ? As the function we are referring to is actually 'y=pi-x' (also can be figured out from the graph itself)

[@DrMathaholic]

Oh yes. While writing I wrote less equal. In this scenario what you said is correct. If it would have been strictly less than pi then we will take average.

[@dimbanimvula3672]

Thank you so much

[@DrMathaholic]

Welcome 😊

[@the.rajkumar.sawant]

Thanks you sir. Keep going

[@DrMathaholic]

Thanks @Transformers Rj.

[@tomtac820]

Sir can we write cosnpi = (-1)^n ? And sir do every time we would have to draw rough graph in order to get continuity of FS ?

[@DrMathaholic]

Yes

[@DrMathaholic]

Its always good to draw graph...it gives you idea . 😊

[@DicksonMayimbo]

Can the concept of old and even functions be used ?

[@DrMathaholic]

Yes..

[@DicksonMayimbo]

Question was in reference to H/w I would also like to inquire on the limits I must consider for Ao and An ,

[@DrMathaholic]

​@DicksonMayimbo yes, it's an even function. Limits will be -pi to pi. Since its an even function u can replace limits by 0 to pi and multiply integrally by 2

[@AhsanAli-op5zb]

when i calculated bn ,its value is different than yours.can you check? ive tried it 3 to 4 times

[@yadneshgujar8991]

Ans of summation 1/n^2 is coming when we are putting pi in fourier series but my doubt is ,how can we put pi if it is not included in interval I mean there is not inequality sighn on pi or -pi given interval at 14:10

[@DrMathaholic]

If I would have written close interval then function is continuous at pi and sum of Fourier series f(pi). But I have put open interval( as u pointed out) so graph will be discontinuous at pi, 3pi and so on. So sum of Fourier series will be average of right and left hand limit at pi, while doing so we get ( pi^2/2+pi^2/2)/2= pi^2/2.

[@robertbat]

SIR AT 9.59 LHL IS PI - X

[@DrMathaholic]

at 3:33 i defined function as x on the interval o to pi. so LHL will be -π- x right???

[@robertbat]

@@DrMathaholic sir I don't understand plz can you explain

[@DrMathaholic]

@@robertbat i meant -π-x... That's what the function is defined on the left side

[@pratikkontamwar7185]

Sir I have doubt on last question which you gave us to hw. My doubt is how x^2/2 is periodic also interval has length of 3pi and also we can find fourier series only for periodic fn ?

[@DrMathaholic]

You only look in the interval -pi to pi. Graph will be of the shape U. Then you draw the same graph in interval pi to 3 pi and so on

[@pratikkontamwar7185]

Sir but you took interval of -pi to 2pi and after 2pi there is no repeatation and interval also has length 3pi so I didn't understand

[@DrMathaholic]

@@pratikkontamwar7185 It - pi < x < pi. Its less than sign, its badly written,i agree, so it's x

[@robertbat]

SIR AT 7.22 a 0 is pi i calculated

[@DrMathaholic]

a_0= integral -pi to 0 (-pi-x) dx + \integral 0 to pi (x) dx =0

[@robertbat]

@@DrMathaholic sir why you take negative pi - x it is pi - x only na ???

[@adeshchoudhar9558]

Sir, at x=pi, RHL would be 0. Is that correct?

[@DrMathaholic]

At what time? You are talking about homework question or other questions which I have solved?

[@adeshchoudhar9558]

@@DrMathaholic 13:25, the question you were solving.

[@DrMathaholic]

@@adeshchoudhar9558 oh yes, it will be 0. Thanks for pointing it out.

[@Taha-en4nx]

For the homework question x=pie/2, right??

[@DrMathaholic]

I have not worked it out. I will solve and confirm. Thank you for posting.

[@DrMathaholic]

Have you tried for x=π?

[@Taha-en4nx]

I was getting pi²/6+ all cos terms

[@Taha-en4nx]

So cos terms converges to zero at pi/2

[@DrMathaholic]

@@Taha-en4nx when we have term cos(nx), then here putting x=π/2 we get cos(nπ/2) and it will take value 0,-1,0,1 and so on..

[@kartikmandhan8850]

At 0:27 ,is the theorem only valid for period 2pi?

[@DrMathaholic]

It is true for any piecewise continuous and periodic function with period 2L. I will upload a lecture which will contain examples over an arbitrary period.

[@DrMathaholic]

Here is the link which have examples:. kzbin.info/www/bejne/bZvSkn-XoJJmp8k

[@pranavpawar7504]

x=pi. But I have a doubt.. while we are sketching the graph should we exclude x=pi , 3pi,5pi.... And x=-pi,-3pi,-5pi ? Are these ppen points and is function defined on that point? IS FUNCTION GRAPH CONTINUOUS AT THAT POINTS?

[@DrMathaholic]

If I would have written close interval then function is continuous at pi and sum of Fourier series f(pi). But I have put open interval( as u pointed out) so graph will be discontinuous at pi, 3pi and so on. So sum of Fourier series will be average of right and left hand limit at pi, while doing so we get ( pi^2/2+pi^2/2)/2= pi^2/2.

[@pranavpawar7504]

Thank u sir😃

[@komalpandya5785]

It's x=pi

[@freakhuehue4456]

You need to take it slow and also explain the each term ....you are going like a bullet train...

[@DrMathaholic]

Oh..thanks for the input. Suggestion: click on settings (wheel) -> playback speed-> decrease the speed as per your convenience