how to get the Fourier series coefficients (fourier series engineering mathematics)

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  • čas přidán 3. 01. 2019
  • Learn how to derive the Fourier series coefficients formulas. Remember, a Fourier series is a series representation of a function with sin(nx) and cos(nx) as its building blocks. Meanwhile, a Taylor series is a series representation of a function with x^n as its building blocks. These are two must-know series in your calculus and engineering math classes.
    Check out the complex Fourier series here: • Complex Fourier Series...
    💪 Support this channel, / blackpenredpen
    ‪@blackpenredpen‬

Komentáře • 336

  • @arequina
    @arequina Před 5 lety +177

    Every time I watch one of your videos, my love for mathematics just keeps increasing. Fourier series was never explained like this in any of my classes. We are just told to accept it.

  • @DasIllu
    @DasIllu Před 5 lety +221

    Points at Fourier and says it's just a name.
    1000000 Engineers feel a cold shiver runing down their spines without knowing why.

  • @davidkwon1872
    @davidkwon1872 Před 3 lety +29

    Thank you.
    Finally I understand what the Fourier is all about.
    Comparing it with Taylor is awesome.
    You are a great teacher.

  • @elenaclaramaria8577
    @elenaclaramaria8577 Před 5 lety +39

    I love you, you’re such a great and humble person man. All the best from Italy, watching you to prepare my Calculus Exam!

  • @connoratkinson8897
    @connoratkinson8897 Před 2 lety +16

    You're a blessing to calc students everywhere thank you so much

  • @andrewhaar2815
    @andrewhaar2815 Před 5 lety +67

    I learned about these recently in my partial differential equations class and I think I can shed some light on why you would multiply by cos(nx) or sin(nx). When Fourier series come up in a PDE the sin and cos terms are eigenfunctions after separating variables. There is a theorem in PDE about Sturm-Liouville differential equations that says that if the DE for the eigenfunctions is in Sturm-Liouville form then the eigenfunctions are orthogonal to each other with a specific weight function, which comes from the form of the DE (and form a complete set). Knowing about orthogonality of functions it would seem only natural to multiply by orthogonal functions. It is just like in linear algebra when you have an orthogonal set you can easily calculate each coefficient in a linear combination using the fact that the dot product is 0 when vectors are orthogonal. One other thing I think is interesting to note is that the coefficient term for a_0 is also the average value of f(x) on the interval. Something interesting and fun to consider is why it would be the average. There are good intuitive reasons...

    • @asxxsss6106
      @asxxsss6106 Před 5 lety

      The term 0 is part of the cosine sum (in reality, the sum goes to 0 to infinite)

    • @andrewhaar2815
      @andrewhaar2815 Před 5 lety

      @@asxxsss6106 It is, but you cannot get a_0 from plugging 0 in for n in the a_n term. Also, as I said, the expression for a_0 is also the average value of f(x) on the interval from -pi to pi, which you can justify intuitively as well as rigorously.

    • @Mayank-mf7xr
      @Mayank-mf7xr Před 4 lety

      this was a very good read . very well written indeed friend

  • @boogychan
    @boogychan Před 2 lety +2

    You are so good Mr. Blackpenredpen . Never learned Fourier series in this way. The way you showed the derivation of the formulas made much sense. Thanks a lot.🥺👍

  • @whatidoknow3417
    @whatidoknow3417 Před rokem +8

    Finally thank you sooo much!! After being pushed around all over CZcams and different materials, I actually found something that relates to what I'm doing and it's then broken down and explained gradually in a way our lecturer didn't bother to do. Thank you soo much. I finally understand have a full grasp of what I'm doing. While watching the video, I was also proving the cases so I can defend everything you've taught. Thank you.

  • @rybaplcaki7267
    @rybaplcaki7267 Před 5 lety +63

    please Fourier transform derivation, complex Fourier or other Fourierr-ish stuff

    • @JPK314
      @JPK314 Před 5 lety +1

      isn't this the fourier transform derivation?

    • @crismal6477
      @crismal6477 Před 4 lety +2

      JPK314 no, this is the Fourier series we are dealing with in this video

  • @emirbakunic2623
    @emirbakunic2623 Před 3 lety +3

    Great videos! You're making student lives so much easier. Best math teacher ever! All the best from Bosnia!

  • @calvinjackson8110
    @calvinjackson8110 Před rokem +6

    Great presentation.
    Have a better understanding of Fourier series now. How this man came up with the series and transform is beyond me.

    • @Sugarman96
      @Sugarman96 Před 6 měsíci

      It's wild to think that he got there because he wanted to solve the heat equation

  • @kristianfella-glanville
    @kristianfella-glanville Před rokem +2

    Great video. You explained it in a really nice (completely perfect I'd say) way and I enjoyed filling in the gaps with the double angle formulas and stuff.

  • @sdsa007
    @sdsa007 Před 3 lety +1

    wow, best work I've seen so far..thanks! wish you can make it longer... I haven't seen any tutorial where they use Taylor series as an analogy, but it helps to have a simpler infinite series as an analogy.

  • @aaronwong7088
    @aaronwong7088 Před 4 lety +1

    i like how each time i don't understand a topic and i saw your videos i heave a sigh of relieve......

  • @ozzyfromspace
    @ozzyfromspace Před 3 lety +20

    17:30, “not [a] simp” - a man of culture, I see 😂😂😂

  • @TheCeava
    @TheCeava Před 3 lety +5

    wow ! you are amazing ! 2 weeks lecture of my prof isnt even comparable to your video ! Thanks so much for this video !! helped me A LOT. also your attitude is amazing :) keep going

  • @Gold161803
    @Gold161803 Před 5 lety +4

    I figured out that part about m being the same as n! The integrands can be written as (cos((n-m)x)+/-cos((n+m)x)/2, and since we've shown twice that the integral from -pi to pi of cos(kx) is zero when k is nonzero, the only thing that matters is when n-m=0, so we can integrate 1/2 over (-pi, pi).
    I know you said you're making a follow-up, I'm just really proud of myself for some reason :)

  • @muazzamhmaiyali251
    @muazzamhmaiyali251 Před 7 měsíci

    This happened to be the first time for me to understand this furrier series. Thank you abundantly, sir.

  • @misugijun
    @misugijun Před 5 lety +2

    that was really good explanation of that building blocks. I never knew it came up like that before. thanks for the derivation. now i feel i understand more on the background of fourier series

    • @blackpenredpen
      @blackpenredpen  Před 5 lety +1

      misugijun thank you!! I use that analogy quiet often. Such as when we solve higher order DE with constant coefficients, e^rt is the building block.

  • @donovancassidy-nolan5553
    @donovancassidy-nolan5553 Před 5 lety +2

    Amazing!!! Beautiful to see where that silly 1/(2*pi) prefactor comes from!!

  • @carlmcgrath484
    @carlmcgrath484 Před 2 lety +1

    Love this guy's channel. never stop holding that mic and saving my degree. big love from England

  • @1albert
    @1albert Před 11 měsíci +1

    Bro you look so happy teaching math, i wish professors and teachers would also have the same passion and love for teaching and solving math problems
    Your energy altered my mood too, i was so stressed
    Thank you so much for everything
    With Love from syria❤

  • @kenankaravoussanos7253
    @kenankaravoussanos7253 Před 5 lety +1

    Literally studying this right now. Thanks for the derivation!

  • @maryamgholinasab4531
    @maryamgholinasab4531 Před 2 lety

    absolutely the best teacher in CZcams thank you.

  • @nurulnurnadirahshafizan5378

    very clear and my curiousity already explained. Thank You Vey Much.. May God bless you

  • @zcl5577
    @zcl5577 Před rokem

    Although I have already learned Fourier series, your explanation gave me new inspiration😀

  • @anagabycano3370
    @anagabycano3370 Před 3 lety +1

    this video actually made me enjoy math! thank you!

  • @_DD_15
    @_DD_15 Před 5 lety +5

    This is getting interesting. Carry on with Laplace as well ☺️

  • @HosRo4161
    @HosRo4161 Před rokem

    Clear and concise! Excellent, thank you!

  • @evazhang3232
    @evazhang3232 Před 2 měsíci

    IT IS SO SATISFYING. THANK YOU!!! I really appreciate you list the first two lines shows the taylor and fourier and proceed everything afterwards. I was always confused that how does the summation of cos and sin come from at the first place. MATH IS BEAUTIFUL! Would you please explain how to get 0 and pi when n not equal and equal to m? orthogonality ? do you have videos on that? Thank you so much!

  • @johnsonisreal4530
    @johnsonisreal4530 Před 3 lety

    Respect brother , definitely you explain all the magic for me.

  • @tutoredwin6119
    @tutoredwin6119 Před 2 lety

    Great teacher, I can now derive Fourier coeffients, thank you so much

  • @qracy-kun5288
    @qracy-kun5288 Před rokem

    this guy is literally became my math teacher
    appreciate your teaching

  • @elpaso4765
    @elpaso4765 Před rokem

    Thank you so much. You explained it really really well.

  • @lal7030
    @lal7030 Před 4 lety

    Oh that's really clear, thanks blackpenredpenbluepen! 😊

  • @RenyxGhoul
    @RenyxGhoul Před 3 lety

    Thank you, I can now answer a similar question but with different constants for sine and cosine

  • @matthewjameswalker721
    @matthewjameswalker721 Před 2 lety

    This is so fun. Great presentation!

  • @oldnordy
    @oldnordy Před 5 lety +4

    bprp,
    You introduce cos(mx) and sin(mx) because that is the proper projection onto the (orthogonal) cos (and sin) function spaces, as are geometric projections to figure out values in geometric space, e.g., Euclidean space.

    • @blackpenredpen
      @blackpenredpen  Před 5 lety +3

      I actually just learned and heard about this from other comments too. Very cool!!! : )

  • @kei3300
    @kei3300 Před 3 lety +1

    I love you so much thank you for being good at teaching math unlike most professors.

    • @calvinjackson8110
      @calvinjackson8110 Před rokem

      You have no idea, NO IDEA, how hard some of those professors work to do the best they can. Everyone is different and is an individual. Your professor may be saying about you that he wish you were like some other student. Would you appreciate such a comparison of your abilities with someone else's??? I think not. Every professor cannot be like this guy just like you cannot be like Einstein!

  • @_DD_15
    @_DD_15 Před 5 lety +2

    Also a 2nd video about non periodic functions would be nice, convergence, when to use Fourier and when Taylor and so on. The Fourier analysis is huge. One of the best parts of Calculus

  • @HannahNimeKiak
    @HannahNimeKiak Před 2 měsíci

    Well explained 👏 Really helpful

  • @joankerubo4943
    @joankerubo4943 Před 2 měsíci

    I really enjoyed watching 😊verry nice teaching

  • @Jacob-uy8ox
    @Jacob-uy8ox Před 5 lety

    You will make my mechanical engineering career easier man, you are the best!

  • @idunablack2592
    @idunablack2592 Před 4 lety

    Your videos are saving my university studies

  • @leecerin7483
    @leecerin7483 Před měsícem +1

    I love how you added the note about mx I was starting to stress about it😂😂😂

  • @SoyFerchow
    @SoyFerchow Před 5 lety +10

    Ey, I love you, saludos desde México. ❤

  • @collinsnjeru5134
    @collinsnjeru5134 Před 3 lety

    Well explained thank you very much... One subscription added

  • @hassanz96
    @hassanz96 Před 2 lety

    The term containing bn confused me a lot. Thank you very much sir for explaining it clearly.

  • @zjyub
    @zjyub Před 4 lety

    You are a total math badass!! Thank you

  • @anmolchaurasia5738
    @anmolchaurasia5738 Před rokem

    I am an Indian🙏🙏 and I really like the way of your explanation ,👍👍
    it was just....wow.....👌👌

  • @eustacenjeru7225
    @eustacenjeru7225 Před 2 lety

    Explained it quite easily

  • @ritesha8050
    @ritesha8050 Před rokem

    thanks, the vids are addictive

  • @maxmoe7244
    @maxmoe7244 Před 3 lety

    Hi love the channel been watching for ages! Just mentioning it seems a bit not-legit swapping between the m-n around for the sum then swapping it back for the final a(subN) constants

  • @anandjaisingh3540
    @anandjaisingh3540 Před 5 lety

    Это очин красивое видео. Спасибо. Really appreciate your hard work

  • @KevinS47
    @KevinS47 Před 5 lety

    BlackpenRedpen, actually, the reason why you should multiply by Cos(mx) or Sin(mx) "f(x)" is because you exploit what's called the "inner product"; I am not 100% sure that this is what Fourier was thinking at the time, but if you know what an innerproduct is, it's simple to see that you have to multiply by a new function to get a specific coefficient. (It's a bit like a dot product except between scalars.. and what do you do when you want to extract the "x" component of a vector? you dot the vector with the unit vector "x", which is sort of what you are doing here basically with cosines and sines)

  • @andrewpappas7198
    @andrewpappas7198 Před 2 lety

    Great explanation!

  • @boussagmanmorad9473
    @boussagmanmorad9473 Před 11 měsíci

    best math teacher ever

  • @zsigmondforianszabo4698

    Amazing video! I watched three videos since, did not get it and this one finally helped me a lot. They jumped over details i couldn't catch because i get things slow and now this explains everything well. Taylor comparison also came handy for a freshman! Really good work!

  • @charlie3k
    @charlie3k Před rokem

    Can you do a video that shows how to go from the Fourier series to the Fourier transform? Thank you so much!

  • @CalculusPhysics
    @CalculusPhysics Před 5 lety

    you should do one on the Fourier Transform as well!! like see where it comes from, maybe similar to 3Blue1Brown’s video but go more in depth! just an idea tho :P

  • @rsssfgr1374
    @rsssfgr1374 Před 2 měsíci

    One 20 minute video later and i get what my lecturers spent hours trying to teach us. Very good video happy smiley zero.

  • @KingRustee
    @KingRustee Před 5 lety +5

    When we learned the Fourier series in my calc class, it was taught for any interval [-p, p] will you be making a video on this?

    • @angelmendez-rivera351
      @angelmendez-rivera351 Před 5 lety +4

      Brendan Russell Well, he said you can also do it for 2π instead of π, but considering that the formulas he gave are inner products, it’s implied that it works for any p.

  • @manuelodabashian
    @manuelodabashian Před 5 lety +3

    After finishing the Fourier analysis start on laurent series

  • @xfcisco
    @xfcisco Před rokem

    the best way to think about this and get a true and deep understanding about it, is if your approach it from a linear algebra point of view. the key to the whole thing is inner products and they will solve all yo' problems

  • @Nemoguzapomnit
    @Nemoguzapomnit Před 5 lety +1

    Oh at last about FS! Thk you sir!

  • @Mau365PP
    @Mau365PP Před 5 lety +5

    Now the complex version with euler's formula 😁

  • @mihaiciorobitca5287
    @mihaiciorobitca5287 Před 5 lety

    It's really easy to calculate the integral bounded by -π and π of cos(nx)*cos(mx) dx ,where n is different by m and also solving the same integral but for the function sin(nx)*sin(mx)
    Yah ,both of them are 0 ;)))

  • @richardfrederick1885
    @richardfrederick1885 Před 3 měsíci

    blackpenredpen, you say you don't know why Fourier multiplied by cos(mpi) and then sin(mpi). I know why, because he had the mathematical insight to visualize they were the correct functions (because he was a genius). You say go ask Fourier (can't , he's dead (ha ha). But keep up the good work, you're an inspiration.

  • @tatjanagobold2810
    @tatjanagobold2810 Před 5 lety +1

    Can't wait for the next video ^^

  • @Wecoc1
    @Wecoc1 Před 5 lety +30

    🙂 + 0 + bm pi
    So sad the second 0 is not as happy as the first

  • @jenny02832
    @jenny02832 Před 2 měsíci

    This is so great ❤

  • @MrConverse
    @MrConverse Před 5 lety

    Can you please do the video showing how the integrals equal π or 0 when m & n are equal or not equal, respectively. I don’t doubt it, I’d like to see it!

  • @grancuadrado
    @grancuadrado Před 5 lety

    Awesome! How about Fourier transform for solving ODE?

  • @tonymcmonster1798
    @tonymcmonster1798 Před 4 lety +4

    Did you miss out a0/2 for a special reason, or just leave as a0 for initial learning simplicity? Thanks for even seeing this (if you do)

    • @boogychan
      @boogychan Před 2 lety

      If you take a0/2 instead of a0 , umm.... a0 was a constant and a0/2 is still a constant. So you can take any number, but it needs to be a constant.😊

  • @sashamuller9743
    @sashamuller9743 Před 4 lety

    thank you for this good video

  • @michaelhunt2222
    @michaelhunt2222 Před 5 lety

    Have you done videos on parametric differentiation and parametric integration?

  • @nrc9275
    @nrc9275 Před 4 lety

    Please make a video on Solution to wave equation in form : £An e ^in2π(x−λt)/L

  • @abdelrahimabdelazim6963

    God bless you
    From Egypt

  • @xKreesherZ
    @xKreesherZ Před 5 lety

    i don't really understand at 14:07, if an is a constant for each term of the summation shouldn't it be like:
    a1+a2/2+a3/3+...+an/n= 1/pi integral of f(x)cos(nx) from -pi to pi?

  • @MM-ck8om
    @MM-ck8om Před 2 lety

    My best lecturer

  • @glennkrafczyk
    @glennkrafczyk Před 2 lety

    Thank you so much!!!!!!

  • @mwaalwajunior8084
    @mwaalwajunior8084 Před 2 lety

    well explained!😊

  • @armpit1648
    @armpit1648 Před 4 lety

    4:20 I don't quite understand the motivation to compare with the taylor series and then decide differentiation vs integration. Why do we need to do differentiate? Why do we need to integrate? What is our end goal with this?

  • @mcNakno
    @mcNakno Před 2 lety

    thanks man, fantastic work, I was pulling out my hair when my Calculus book didn't explain how to compute a_0

  • @ozzyfromspace
    @ozzyfromspace Před 4 lety

    I subbed and went to my second video. The number went from 379k to 380k subscribers, I feel proud!

  • @ojoprecious1252
    @ojoprecious1252 Před rokem

    Thank you! ❤️

  • @loveen3186
    @loveen3186 Před 2 lety

    thanks man, amazing

  • @CazoDK
    @CazoDK Před 5 lety

    Hey BPRP. Will you elaborate on the fourier transform and perhaps its relation to quantum mechanics? I'm specifically talking about Heisenberg's uncertainty principle that you could use as an example of such a transform. Keep up the good work!

    • @angelmendez-rivera351
      @angelmendez-rivera351 Před 5 lety +1

      Cazo He is not a physicist. I doubt he is trained to talk about subjects of quantum mechanics. He is a mathematician, or a mathematics professor, anyway.

    • @blackpenredpen
      @blackpenredpen  Před 5 lety

      Yea, just like Angel said. I actually don't have much knowledge in quantum mechanics. But yea, I like to solve math problems. : )

    • @CazoDK
      @CazoDK Před 5 lety

      blackpenredpen That’s okay. I still love watching your videos anyway, of course. I’ve learned a lot about Fourier series because of you, so thank you!

  • @gurdombajo2881
    @gurdombajo2881 Před 4 lety

    Please do one on application of Fourier series

  • @cryoine7194
    @cryoine7194 Před 11 měsíci

    I'll always say it 0 has got to be my favorite number, it just makes things easier

  • @mahjoubahmed1287
    @mahjoubahmed1287 Před 3 lety

    Thank you very much

  • @nawarajsubedi4388
    @nawarajsubedi4388 Před 4 lety

    Great video..
    Love frn Nepal.

  • @eustacenjeru7225
    @eustacenjeru7225 Před 2 lety

    Great work

  • @encounter_life3403
    @encounter_life3403 Před 3 lety

    Thank you!

  • @alexismisselyn3916
    @alexismisselyn3916 Před 4 lety

    Do the terms also become smaller as n gets bigger? In taylor you can have a decent approximation with a few terms, do fourier expansions work the same? With maybe a little-o term?

    • @vaughanwilliamson173
      @vaughanwilliamson173 Před 4 lety

      The terms don't necessarily get smaller as n increases. You can be very creative. Consider the uses of the Fourier series. One of the classic Fourier decompositions is that of the square wave. The square wave has no even valued harmonics (multiples of the fundamental frequency). So there is no second harmonic, but there is a third, no fourth, but a fifth and so on. So there is no sin(2x), nor sin(4x), nor sin(6x), nor any cosines. Only sin(x), sin(3x), sin(5x) and so on exist in the square wave decomposition with perhaps a constant offset if the average value is not zero.
      Take another waveform other than the square wave and there is any possibility that say the 5th harmonic is greater than the 3rd. You can have as much fun as you like.

  • @jayjayf9699
    @jayjayf9699 Před 4 lety +1

    If cos(nx) is a even function how come on the interval of -pi and pi it equals zero ? 8:53

  • @maxdemuynck9850
    @maxdemuynck9850 Před 3 lety

    great video!

  • @kianushmaleki
    @kianushmaleki Před rokem

    amazing . love it

  • @surajthakur4839
    @surajthakur4839 Před rokem

    Thank you sir form India 🇮🇳