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Parseval's Identity, Fourier Series, and Solving this Classic Pi Formula
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- Опубликовано: 7 авг 2024
- To celebrate #PiDay we solve the Basel Problem - that the sum of reciprocals of square naturals is pi^2/6 - using techniques from Fourier Analysis, in particular Parseval's Identity, which is a sort of infinite dimensional analog of Pythagoras.
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0:00 The Basel Problem
1:06 Fourier Series Refresher
3:22 Parseval's Identity
5:13 Inner Products & Generalized Pythagoras
9:46 The proof that n^2/6=1/1+1/4+1/9...
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Parseval's identity is very powerful! It can also solve the much less well-known sum of 1/(n^2+1) from -infinity to infinity. Using the Fourier series of f(x) = e^x (-1 < x < 1), we can find that the answer is π coth(π) = 3.1533, slightly greater than the integral of 1/(x^2+1) over the same interval, which of course is just π. Happy π day!
This proof of the Basel problem is mind-blowing. I had to prove that result for zeta(2) as part of an MSc thesis and it was a hard job. My problem now is that I don't understand Fourier series very well. I think I'll do some work in that area to improve my understanding. There must be a lot of proofs for zeta(2) =pi^2/6; I've seen 3 or 4 proofs on RUclips, but, of course, this is the simplest one I have ever seen.
I'm currently taking math 346 at UVic, how did I not know you had a playlist about Fourier series?!? Thankful I found this before my next midterm 😅
You do not know so many things. Don't come here asking others why.
Great video as always, thank you! By the way, another way to evaluate zeta(2) which doesn't rely on Parseval's theorem but does use Fourier series is simply to use the Fourier series of f(x)=|x|. See what happens!
Sir please continue uploading videos.. these are really helpful
Never knew this powerful approximation existed! Happy belated pi day ;)
Love everything you do and thank you so much for all these videos from Calc I to Fourier Seties. And it has helped me throughout my engineering course. But I think it would be nice to add Euler's exponential version and Fourier Ananlysis in this series.
Super nice! Thanks!
Oh my god, that was incredibly straightforward! 😨
It was great. Thank you.👌
I like your educational videos fully knowledgeable
Thank you. Really enjoyed. This series could be further developed to include more of the topic and applications in various areas, such as engineering, economics, finance, etc.
awesome!
Great video for the pi day! Well done, Trefor.
Thanks!
please tell me more about parsevals thm. and pretty much everything in this vid...
omg its so helpful
Great vid. I would love a small video on the harmonics of Fourier series.
Wow so interesting
Love the shirt!
Awesome
At 10:26,why do I get the coefficient of the Sin(nx) as bn,not an
Even , even if you decide to don't learn and don't enjoy, you can't!
You don't have any choice and it's not optional! you'll learn and enjoy!
Great video.
Thank you professor
haha, thanks!
@@DrTrefor I like this but how would you use this to estimate pi if you use pi for the limits of integration? I guess because it’s a symmetric function you could finagle it so the length of the interval of integration didn’t matter keeping in mind that cosine integrates to an odd function and sine integrates to an even one.
@@dominicellis1867 The idea is to integrate a periodic function for 1 period, so, in cases of trigonometric functions, an interval of 2 Pi, which results in zero.
Hi professor, for a homework assignment I need to calculate what the infinite sum of 1/n^4 converges to using Plancherel's theorem and the fourier series of f(x) = x^2 from -pi to pi. Everything I find on the internet is about Parseval's theorem though which I'm not allowed to use. When I try to use Plancherel's theorem I get 7pi^4/180, but the right answer is pi^4/90. Do you have any suggestions? Thank you professor
love that
awesome
What a fun little alternative derivation! Great Pi Day post!
"In this vide-you " looool every thing
So,
how do you like your π : à la mode or plain ?
Really cool :D
really well explained as well
understanding Fourier series in the context of vector spaces feels much more natural, especially with Parseval's identity. Looks like a mish-mash of symbols at first glance but through the lens of linear algebra it just clicks in my head why the identity looks that way
Great math from a Knight of the Round Table! Just joking, excellent video as always :D
Hi Professor, any plans for a future math series? I think it would be cool if you make a new math series, since your explanations are always excellent. An Algebra series maybe? Anyways nice video as always
Definitely want to do an algebra series at some point
Good
love that shirt
The division by π in front on the integral comes from integrating the sinus, and the cosinus over an even interval right? Since it is an constant value, you can get it out of the summation, and then I guess you divide everything by π. But why is the constant a0 not divided by π then?
Also the definition of inner product that we use at university does not contain a division by π?
Happy Pi Day!
Happy Pi day, everyone. I hope you make your favorite kind of pie (which includes pizza since the word means 'pie' in Italian) and also at the same time measure the diameter of your pie and its circumference to derive a close approximation of pi.
Pizza non significa pie in italiano mmerricano
hmmm... imagine if we can pick a periodic function whose value at a point corresponds to whatever number we want (euler's number e, golden ratio, euler-mascheroni constant, etc) maybe we can write those numbers as series, using function expansions as well
Good day Dr. Trefor, At about 3:19, maybe you meant a(subn) and b(subn) instead of a(subm) and b(subm), if you look at the top formula? I'm a bit confused. Great and clear presentation as always, Jan-W
Oh yes! Sometimes written with m and sometimes with n, doesn’t matter which as long as one doesn’t interchange them ha!
@@DrTrefor Dr. Trefor, thank you for your reply, it was only meant for the record.This did not affect the clarity of your lecture for me. Thank you for all your other presentations... Greetings from Holland, Jan-W
Sir here i guess bn= (2*pi*(-1)^n+1) / (n^2)
Please check and reply sir
If you find it interesting enough, could you make a video on the Gibbs phenomenon; other explanations that I saw are just to confusing (unlike the ones I see here)
I actually talked about this earlier in the Fourier Series playlist!
@@DrTrefor Oh...ok thanks, that's good to hear. I'll check it out
sines and cosines act as orthogonal basis to periodic functions? sounds like linear algebra
Indeed, strong parallel to linear algebra throughout this
It sounds like linear algebra because it is linear algebra. The functions R -> R with period T form a vector space of countably infinite dimension, and the standard basis for that vector space is the comprised of sines and cosines. Finding the Fourier coefficients is just decomposing a given vector in terms of that basis.
Furthermore, such a vector space is a inner product space over the real numbers, as described in the video. The basis described is orthogonal with respect to the given inner product. The inner product space is known as L^2 space. It is an extremely important type of inner product space.
@@angelmendez-rivera351 ohhh that makes so much sense... i also heard of special polynomials (legendre and hermite polynomials? there got to be others) being orthogonal bases (maybe for another space?). So if i'm a physicist coming up with a linear equation, and i can prove that the solutions form a vector space, i can do whatever is in linear algebra to probe the solutions further? neat
@@GeoffryGifari The ring of polynomial functions with real coefficients forms a vector space over the field of real numbers R. The Legendre polynomials form a basis of this vector space, and this basis is orthonormal with respect to the L^2 inner product demonstrated here in the video.
Equations in physics are comprised of linear operators acting on vectors from some predetermined, known vector space, and what this vector space is depends on the physical theory you are working with. Physical theories come with physical restrictions on mathematical structures and with boundary conditions. These will determine the types of linear operators permitted by the theory, and the shared vector space they act on. The solutions to an equation comprised of these operators will thus always belong to that vector space, they may not necessarily span the entire vector space. Solving these equations is equivalent to finding the null space of the characteristic operator for the equation, and the null space will always be a subspace of the physically permitted vector space, and since you can always find a spanning set for this null space under suitable conditions, you can always find a basis for this null space. If you have an inner product, which in physics, you typically do, then you can orthonormalize that basis with respect to that inner product.
Another easy way to solve the Basel problem is as follows : Expand the function f[x]=x*(Pi - x) symmetrically in a Fourier Series to get
f[x]= Pi^2/6 -cos2x]-1/4 cos[4x]-1/9 cos6x- ..... For x= 0 you have the solution of the Basel problem .
You have the same tone as a pastor when talking about mathematics
haha I can't decide if this is a good thing or not:D
@@DrTrefor It would be a bad thing however I don't think you sound like a pastor. You sound like this one English teacher that I had when I was in high school.
It bothered me that every time he said "triangle," he meant "right triangle." And he didn't put the right angle indication on the diagram when it mattered. Although it's neat that there was one on his shirt.
Please make video on Riemann hypothesis 🙏
that would be a BIG video:D
@@DrTrefor sure would
You're using the same notation for vectors and inner products. Might be confusing for someone seeing these the first time.
v= and also
✌️ ???????
Low rate of convergence
Yup, not the current series used for computing large values of pi today
@@DrTrefor billion terms for a good approximation
14/3 jijiji
Happy Pi day to everyone!!
Happy Pi Day
ζ(2)=π²/6
Riemann hypothesis 😁👍
@@numberandfacts6174 That has nothing to do with the Riemann hypothesis.
This video is very irrational