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 • 335

  • @arequina
    @arequina Před 5 lety +180

    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 +222

    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

  • @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.

  • @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

  • @boogychan
    @boogychan Před 3 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.🥺👍

  • @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 5 lety

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

  • @ozzyfromspace
    @ozzyfromspace Před 3 lety +20

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

  • @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!

  • @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

  • @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......

  • @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.

  • @muazzamhmaiyali251
    @muazzamhmaiyali251 Před 7 měsíci

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

  • @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

  • @maryamgholinasab4531
    @maryamgholinasab4531 Před 2 lety

    absolutely the best teacher in CZcams thank you.

  • @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.

  • @kenankaravoussanos7253
    @kenankaravoussanos7253 Před 5 lety +1

    Literally studying this right now. Thanks for the derivation!

  • @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

  • @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 :)

  • @anagabycano3370
    @anagabycano3370 Před 3 lety +1

    this video actually made me enjoy math! thank you!

  • @1albert
    @1albert Před rokem +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❤

  • @HosRo4161
    @HosRo4161 Před rokem

    Clear and concise! Excellent, thank you!

  • @rsssfgr1374
    @rsssfgr1374 Před 3 měsíci

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

  • @qracy-kun5288
    @qracy-kun5288 Před rokem

    this guy is literally became my math teacher
    appreciate your teaching

  • @_DD_15
    @_DD_15 Před 5 lety +5

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

  • @lal7030
    @lal7030 Před 4 lety

    Oh that's really clear, thanks blackpenredpenbluepen! 😊

  • @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

  • @zcl5577
    @zcl5577 Před rokem

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

  • @nurulnurnadirahshafizan5378

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

  • @matthewjameswalker721
    @matthewjameswalker721 Před 2 lety

    This is so fun. Great presentation!

  • @HannahNimeKiak
    @HannahNimeKiak Před 2 měsíci

    Well explained 👏 Really helpful

  • @elpaso4765
    @elpaso4765 Před rokem

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

  • @boussagmanmorad9473
    @boussagmanmorad9473 Před 11 měsíci

    best math teacher ever

  • @eustacenjeru7225
    @eustacenjeru7225 Před 2 lety

    Explained it quite easily

  • @SoyFerchow
    @SoyFerchow Před 5 lety +10

    Ey, I love you, saludos desde México. ❤

  • @idunablack2592
    @idunablack2592 Před 4 lety

    Your videos are saving my university studies

  • @joankerubo4943
    @joankerubo4943 Před 2 měsíci

    I really enjoyed watching 😊verry nice teaching

  • @ritesha8050
    @ritesha8050 Před rokem

    thanks, the vids are addictive

  • @zjyub
    @zjyub Před 4 lety

    You are a total math badass!! Thank you

  • @leecerin7483
    @leecerin7483 Před 2 měsíci +1

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

  • @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!

  • @jenny02832
    @jenny02832 Před 2 měsíci

    This is so great ❤

  • @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.

  • @andrewpappas7198
    @andrewpappas7198 Před 2 lety

    Great explanation!

  • @RenyxGhoul
    @RenyxGhoul Před 3 lety

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

  • @tatjanagobold2810
    @tatjanagobold2810 Před 5 lety +1

    Can't wait for the next video ^^

  • @evazhang3232
    @evazhang3232 Před 3 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!

  • @Nemoguzapomnit
    @Nemoguzapomnit Před 5 lety +1

    Oh at last about FS! Thk you sir!

  • @loveen3186
    @loveen3186 Před 2 lety

    thanks man, amazing

  • @sachietkapur
    @sachietkapur Před 5 lety

    FEELS SO GOOD

  • @eustacenjeru7225
    @eustacenjeru7225 Před 2 lety

    Great work

  • @collinsnjeru5134
    @collinsnjeru5134 Před 3 lety

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

  • @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!

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

    My best lecturer

  • @maxdemuynck9850
    @maxdemuynck9850 Před 3 lety

    great video!

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

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

  • @sashamuller9743
    @sashamuller9743 Před 4 lety

    thank you for this good video

  • @ojoprecious1252
    @ojoprecious1252 Před rokem

    Thank you! ❤️

  • @glennkrafczyk
    @glennkrafczyk Před 2 lety

    Thank you so much!!!!!!

  • @joaoarruda138
    @joaoarruda138 Před 2 měsíci

    good work!

  • @encounter_life3403
    @encounter_life3403 Před 3 lety

    Thank you!

  • @khabdullah2694
    @khabdullah2694 Před rokem

    Best explanation 😍😍😍😍

  • @abdelrahimabdelazim6963

    God bless you
    From Egypt

  • @mwaalwajunior8084
    @mwaalwajunior8084 Před 2 lety

    well explained!😊

  • @nawarajsubedi4388
    @nawarajsubedi4388 Před 4 lety

    Great video..
    Love frn Nepal.

  • @anmolchaurasia5738
    @anmolchaurasia5738 Před rokem

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

  • @_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

  • @anandjaisingh3540
    @anandjaisingh3540 Před 5 lety

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

  • @mahjoubahmed1287
    @mahjoubahmed1287 Před 3 lety

    Thank you very much

  • @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!!! : )

  • @kianushmaleki
    @kianushmaleki Před rokem

    amazing . love it

  • @sirkelvinmalunga
    @sirkelvinmalunga Před 8 měsíci

    youre the best!!!!!!

  • @AnantoYusufW
    @AnantoYusufW Před 3 lety

    thank you so much,

  • @michaelhunt2222
    @michaelhunt2222 Před 5 lety

    Have you done videos on parametric differentiation and parametric integration?

  • @zzz15432
    @zzz15432 Před 2 lety

    You are a lifesaver ily 🤍

  • @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

  • @Mau365PP
    @Mau365PP Před 5 lety +5

    Now the complex version with euler's formula 😁

  • @geosalatast5715
    @geosalatast5715 Před 4 lety

    Haha you are the bob ross of maths! Greets from Greece :)

  • @hassanz96
    @hassanz96 Před 2 lety

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

  • @Wecoc1
    @Wecoc1 Před 5 lety +30

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

  • @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

  • @inj1979
    @inj1979 Před 2 lety

    Thank you. 🍎

  • @surajthakur4839
    @surajthakur4839 Před rokem

    Thank you sir form India 🇮🇳

  • @openway8949
    @openway8949 Před 4 lety

    thank you

  • @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

  • @abhisheknadargi7018
    @abhisheknadargi7018 Před 5 lety

    U make great videos 👍😘

  • @Chris-rb8ox
    @Chris-rb8ox Před 3 lety

    Thanks :)

  • @grancuadrado
    @grancuadrado Před 5 lety

    Awesome! How about Fourier transform for solving ODE?

  • @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.

  • @mthokozisindlovu873
    @mthokozisindlovu873 Před 2 lety

    youre the best

  • @cryoine7194
    @cryoine7194 Před 11 měsíci

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

  • @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)

  • @ozzyfromspace
    @ozzyfromspace Před 4 lety

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