how to get the Fourier series coefficients (fourier series engineering mathematics)
Vložit
- 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
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.
I am glad to hear that you enjoy my videos! thank you!
🤣
Points at Fourier and says it's just a name.
1000000 Engineers feel a cold shiver runing down their spines without knowing why.
LOL exactly
Lol I didn't shiver 😀
Thank you.
Finally I understand what the Fourier is all about.
Comparing it with Taylor is awesome.
You are a great teacher.
I love you, you’re such a great and humble person man. All the best from Italy, watching you to prepare my Calculus Exam!
Elena Clara Maria thank you!!
You're a blessing to calc students everywhere thank you so much
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...
The term 0 is part of the cosine sum (in reality, the sum goes to 0 to infinite)
@@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.
this was a very good read . very well written indeed friend
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.🥺👍
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.
I am happy to hear. Thank you!
please Fourier transform derivation, complex Fourier or other Fourierr-ish stuff
isn't this the fourier transform derivation?
JPK314 no, this is the Fourier series we are dealing with in this video
Great videos! You're making student lives so much easier. Best math teacher ever! All the best from Bosnia!
Great presentation.
Have a better understanding of Fourier series now. How this man came up with the series and transform is beyond me.
It's wild to think that he got there because he wanted to solve the heat equation
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.
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.
i like how each time i don't understand a topic and i saw your videos i heave a sigh of relieve......
17:30, “not [a] simp” - a man of culture, I see 😂😂😂
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
Glad to hear. Thanks.
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 :)
This happened to be the first time for me to understand this furrier series. Thank you abundantly, sir.
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
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.
Amazing!!! Beautiful to see where that silly 1/(2*pi) prefactor comes from!!
Love this guy's channel. never stop holding that mic and saving my degree. big love from England
Glad to hear. Thanks.
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❤
Literally studying this right now. Thanks for the derivation!
absolutely the best teacher in CZcams thank you.
very clear and my curiousity already explained. Thank You Vey Much.. May God bless you
Although I have already learned Fourier series, your explanation gave me new inspiration😀
this video actually made me enjoy math! thank you!
This is getting interesting. Carry on with Laplace as well ☺️
Clear and concise! Excellent, thank you!
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!
Respect brother , definitely you explain all the magic for me.
Great teacher, I can now derive Fourier coeffients, thank you so much
this guy is literally became my math teacher
appreciate your teaching
Thank you so much. You explained it really really well.
Oh that's really clear, thanks blackpenredpenbluepen! 😊
Thank you, I can now answer a similar question but with different constants for sine and cosine
This is so fun. Great presentation!
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.
I actually just learned and heard about this from other comments too. Very cool!!! : )
I love you so much thank you for being good at teaching math unlike most professors.
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!
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
Well explained 👏 Really helpful
I really enjoyed watching 😊verry nice teaching
You will make my mechanical engineering career easier man, you are the best!
: )))
@@blackpenredpen what about doing Fourier transform video? :D
Your videos are saving my university studies
I love how you added the note about mx I was starting to stress about it😂😂😂
Ey, I love you, saludos desde México. ❤
Well explained thank you very much... One subscription added
The term containing bn confused me a lot. Thank you very much sir for explaining it clearly.
You are a total math badass!! Thank you
I am an Indian🙏🙏 and I really like the way of your explanation ,👍👍
it was just....wow.....👌👌
Explained it quite easily
thanks, the vids are addictive
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
Это очин красивое видео. Спасибо. Really appreciate your hard work
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)
Great explanation!
best math teacher ever
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!
Can you do a video that shows how to go from the Fourier series to the Fourier transform? Thank you so much!
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
One 20 minute video later and i get what my lecturers spent hours trying to teach us. Very good video happy smiley zero.
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?
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.
After finishing the Fourier analysis start on laurent series
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
Oh at last about FS! Thk you sir!
: )))))
Now the complex version with euler's formula 😁
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 ;)))
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.
Can't wait for the next video ^^
: ))))
🙂 + 0 + bm pi
So sad the second 0 is not as happy as the first
This is so great ❤
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!
Awesome! How about Fourier transform for solving ODE?
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)
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.😊
thank you for this good video
Have you done videos on parametric differentiation and parametric integration?
Please make a video on Solution to wave equation in form : £An e ^in2π(x−λt)/L
God bless you
From Egypt
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?
My best lecturer
Thank you so much!!!!!!
well explained!😊
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?
thanks man, fantastic work, I was pulling out my hair when my Calculus book didn't explain how to compute a_0
I subbed and went to my second video. The number went from 379k to 380k subscribers, I feel proud!
Thank you! ❤️
thanks man, amazing
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!
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.
Yea, just like Angel said. I actually don't have much knowledge in quantum mechanics. But yea, I like to solve math problems. : )
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!
Please do one on application of Fourier series
I'll always say it 0 has got to be my favorite number, it just makes things easier
Thank you very much
Great video..
Love frn Nepal.
Great work
Thank you!
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?
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.
If cos(nx) is a even function how come on the interval of -pi and pi it equals zero ? 8:53
great video!
amazing . love it
Thank you sir form India 🇮🇳