how to get the Fourier series coefficients (fourier series engineering mathematics)
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- Опубликовано: 23 окт 2024
- 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...
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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
Thank you.
Finally I understand what the Fourier is all about.
Comparing it with Taylor is awesome.
You are a great teacher.
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
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
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
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
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.🥺👍
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 :)
i like how each time i don't understand a topic and i saw your videos i heave a sigh of relieve......
Finally thank you sooo much!! After being pushed around all over RUclips 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!
This happened to be the first time for me to understand this furrier series. Thank you abundantly, sir.
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.
Great videos! You're making student lives so much easier. Best math teacher ever! All the best from Bosnia!
Love this guy's channel. never stop holding that mic and saving my degree. big love from England
Glad to hear. Thanks.
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.
17:30, “not [a] simp” - a man of culture, I see 😂😂😂
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❤
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.
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!!! : )
this guy is literally became my math teacher
appreciate your teaching
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!
Amazing!!! Beautiful to see where that silly 1/(2*pi) prefactor comes from!!
I watched all your proof videos. They were awesome. I liked them so much I got hooked on proofs.
Now I will watch your Fourier series and FFT videos if they exist.
Thank u
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.
Although I have already learned Fourier series, your explanation gave me new inspiration😀
absolutely the best teacher in RUclips thank you.
Literally studying this right now. Thanks for the derivation!
I love how you added the note about mx I was starting to stress about it😂😂😂
Clear and concise! Excellent, thank you!
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)
That's a very elaborate explanation thanks😊
This is getting interesting. Carry on with Laplace as well ☺️
very clear and my curiousity already explained. Thank You Vey Much.. May God bless you
I am an Indian🙏🙏 and I really like the way of your explanation ,👍👍
it was just....wow.....👌👌
Great teacher, I can now derive Fourier coeffients, thank you so much
this video actually made me enjoy math! thank you!
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
best math teacher ever
The reason for the difference between the integral of cos(nx)*cos(mx) and sin(nx)*sin(mx) where n = m and n =/= m happens because because if n = m, cos(nx)*cos(mx) and sin(nx)*sin(mx), due to being equal to cos²(mx) and sin²(mx), respectively, are always positve whereas cos(nx)*cos(mx) and sin(nx)*sin(mx) can have negative values.
Alternatively, you can use the trigonometic identity with the product of two angles for n =/= m. The integral you get is:
I(x) = 1/2[sin(nx+mx)/(n+m) - sin(nx - mx)/(n - m)]
which equals from -π to π to 0 because n and m are natural numbers/positive integers and with c being any integer, sin(c*π) = 0.
In fact, you can use the same integral and use n = m, assuming m approaches n because part of the integral results in a 0/0 situation (more specifically, lim n -> m of sin(nx - mx)/(n - m) = x) and I(π) - I(-π) = [π - (-π)]/2 = π. Of course, you can calculate it classicaly with the power reduction for n = m.
MarioFanGamer Well, he sort of explained that in the video already, but I think it’s more illuminating to talk about what it means: it means the functions are orthogonal.
I'll always say it 0 has got to be my favorite number, it just makes things easier
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.
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!
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.😊
Yo! Supp bro. Are these formulas street legal? Can i end up in joint if i use them in public? Thanks!
After finishing the Fourier analysis start on laurent series
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!
Your videos are saving my university studies
You will make my mechanical engineering career easier man, you are the best!
: )))
@@blackpenredpen what about doing Fourier transform video? :D
Thanks from India 😊🇮🇳
4yay! Series! Also, I have a number theory challenge for you. Consider the pair of numbers n - k and k + 1, where n is a natural number and k = 0, 1, ..., n - 1. For what values of n are n - k and k + 1 co-prime (relatively prime, they share no common factors) for all k? In other words, for what values of n is gcd(n - k, k + 1) = 1 for all k satisfying the conditions I established? Good luck with this :)
Also, I would like to add to the video the fact that if we instead present the series with a0/2 instead of a0, then we get to present the definition of a0 instead as dividing the integral by π, leaving a symmetry between all the coefficients in that they all divide some integral by π. Also, you can form a Fourier series for any arbitrary interval [-L, L]. Sometimes, for very nice functions, one is even able to let lim L -> ♾ and give an infinitely converging series.
Also, this has profound implications from linear algebra. What this means is that the set of sines and cosines with all integer frequencies is a functionally complete orthogonal basis for the vector space. It is very nice. This is why we can do this in the first place, and this moreover reflects the form of the coefficients as being inner products.
One final note is that I find interesting how each different series helps in the hierarchy of building functions by using infinite series, allowing us to build more and more functions. Arithmetic allows us to build monomials and binomials. Iteration of this helps us build polynomials. Then we can use infinite polynomials to build smooth functions, which means we can build any sinusoidal trigonometric function. Now we have a series on sinusoidal functions from which we can build even more functions than we can from the Taylor series, allowing us to build step functions, indicator functions, and other waves which are not smooth nor continuous. Then we can form series on those new functions we built to build all sorts of pathological functions, yet allowing us to work with them as if they were not pathological. It also helps lay the foundation for Lebesgue integration. One series allows us to build a basis that we can use to form a series that allows us to build a larger space of functions. Etc. A very useful and interesting hierarchy that I consider to be a happy surprise.
Hamish Blair Correct. Another way to demonstrate it is by noting that, if m = n + 1, then (n - k)/(k + 1) = [m - (k + 1)]/(k + 1). Now, we take the prime factorization of k + 1, and notice that if m shares any factors with k + 1, then the fraction can be reduced easily. But since k + 1 ranges over every number from 1 to k, there is at least one k + 1 that will share one prime factor with m UNLESS m is prime. This confirms that n = p - 1.
A little note though, on your comment, it should read that gcd(p - (k + 1), k + 1) = 1, not 0, because 0 can never be a factor of a nonzero number. I assume it’s a small typo, but yeah.
You can't build any smooth function from Taylor's series, you can only build analytic ones
will newman Any analytic function is necessarily smooth, so yes, you can build smooth functions. Have you studied analysis?
Gcd(n-k, k+1) = gcd (n-k+(k+1),k+1) = gcd (n+1,k+1) =1.
If n+1 happens to be a prime number, then it is true. Otherwise, considering any k, which is 1 less, than any prime factor of n+1 leads to contradiction
I would suggest getting a keyboard that allowes you to do ˢᵘᵖᵉʳˢᶜʳⁱᵖᵗˢ and ₛᵤbscripts. It would make things like a0 to a₀. Also things like lim ₓ→₁f(x).
🙂 + 0 + bm pi
So sad the second 0 is not as happy as the first
Explained it quite easily
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
I would prefer to use:
f(x) = sum_{n=-inf...inf} a_n e^(inx)
#EulersFormula makes it much more concise and, moreover, can be used to _derive_ the other expression with sin/cos from this one and vice versa, thus showing their equivalence. (NB: there might be some weird, subtle real-analytic arguments around absolute vs. conditional convergence in some cases that may or may not cause a problem in at least "bad" cases, but I'm not sure about this.)
That said, if you want to use real sine/cosine only, I'd also prefer
f(x) = sum_{n=0...inf} a_n sin(nx + phi_n)
which more closely illustrates that the point of the series is you are summing waves with both different amplitudes a_n and different _phases_ phi_n. Again, equivalence of these expressions is similarly proveable (namely just note that the sin/cos series effectively just takes "half", so to speak, of the phase factors as tau/4 , or 90 degrees.). Moreover, it transparently lets you see that a square and triangular wave are pretty much the same thing, just depending on where you put the phase. One may also be tempted to explore the ominous world of
"f"(x) "=" sum_{n=0...inf} sin(nx)
subtleetiees concealing #DREAD awwwaaiiitttt..... w000000000000hhhhhhhh.......................................................................
mike4ty4 The fact that this is an additive combination of waves is already obvious, even without invoking the difference in phase. I do agree that the complex form is better, but for students who are learning about, this method may be more convenient regardless because in many courses where they teach this they do not allow you to use any complex algebra or complex analysis whatsoever. Also, although that formula is indeed more concise, this illustrates the meaning of it better, especially when understanding the linear algebra behind it.
One 20 minute video later and i get what my lecturers spent hours trying to teach us. Very good video happy smiley zero.
Please do one on application of Fourier series
Now the complex version with euler's formula 😁
Hello. I don't know if you'll answer this but I would like to know if it is ao or ao/2. I have been confused about that. I don't know the correct one. Please help
It is a0 when u calculate the a0 using 1/2pi *integral and it is a0/2 when calculating using 1/pi*integral!
Just found that out today 😂
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.
My best lecturer
Respect brother , definitely you explain all the magic for me.
thank you
Does EVERY function converge in the interval (-pi,pi) using the Fourier Series?
And what happens if you change the interval? For example (-3pi,3pi), will it converge too?
I have the same question. Do you have to integrate it from the beginning by yourself? Or is there any general formula for whichever interval you choose? If anyone knows, please let me know
@@turel528 Not every function will converge. For instance, the Fourier series of tangent of x will have trouble, because the integrals will diverge nearby the asymptotes.
You can change the interval to anything you want, as long as it fully captures a complete period of the periodic function. For instance, for a function of the same period as the original sine or cosine, I'd use -pi to pi as the interval. Whereas for a function with a period of 6*pi, I'd use -3*pi to + 3*pi.
The term containing bn confused me a lot. Thank you very much sir for explaining it clearly.
Shouldn`t that be yay^4 ?
Nah, 4yay makes more sense given the pronunciation. In other words, 4ay^2
go 4yay!
God bless you
From Egypt
Can you do a video that shows how to go from the Fourier series to the Fourier transform? Thank you so much!
Thank you so much. You explained it really really well.
Thank you, I can now answer a similar question but with different constants for sine and cosine
thanks man, fantastic work, I was pulling out my hair when my Calculus book didn't explain how to compute a_0
Thank you sir form India 🇮🇳
I subbed and went to my second video. The number went from 379k to 380k subscribers, I feel proud!
Oh that's really clear, thanks blackpenredpenbluepen! 😊
Do a video on Laurent series.
Commander Shepard I approve
Please post more Laplace and Fourier series stuff
Well explained 👏 Really helpful
I really enjoyed watching 😊verry nice teaching
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!
Oh at last about FS! Thk you sir!
: )))))
King of Fourier.
If cos(nx) is a even function how come on the interval of -pi and pi it equals zero ? 8:53
Ey, I love you, saludos desde México. ❤
Why can you multiply by cos(mx) inside each integral's terms ?
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?
This is so fun. Great presentation!
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 ;)))
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
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?
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!
this confuses me. at 9:09 you say "-1/n * sin(n-pi) = 0" but punching it into desmos, it most definitely does not equal 0 for integer n.
I have 15 years temperature data. I want to remove seasonality variation from the time series data, so how can I get the constant term ao, the amplitude and the phase change by using these given temperature data? thank you
is the formula for a0 = 1/2p all the time?
in mutiply both side of cos(mx) or sin(mx) where's sigma going?
Awesome! How about Fourier transform for solving ODE?
Prof I really love huh......as how as I am I have been struggling with this Fourier but thank Goodness 😅 I came across this I am going to be proving this in my exam in two days time........but there is a big problem I try to confirm what u said if n=m with real figure in the integration I got zero instead of pie that u said.....please I need a quick response thanks 🙏 I am cravingly in love with ur teaching
since an and bn are fxns, is it really okay to say that int an cos(nx) =0 ? you also swapped the order of the sum sign and the integral which can only be done if the sum/integral is uniformly continuous (I think-don't quote me), but never proved that.
I have seen somewhere DC term is taken as ao and somewhere ao/2. Can you tell please why?