Complex Fourier Series

This video will describe how the Fourier Series can be written efficiently in complex variables.

These lectures follow Chapter 2 from:
"Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz

This video was produced at the University of Washington

[fnegnilr]

I like how you are using geometric figures to keep us grounded in the explanations.

[thomasdehaeze]

Minor comment about the integral : if j=k, then the exponential in the integral is equal to one, therefore the integral is equal to 2pi. Then we don't have to deal with limits.

[christophs]

I like this video series a lot. Very energetic and enthousiastic style of presenting! One thing i want to mention is that this only proves the orthogonality part, it does not show that the psi-functions form a basis.

[simong]

g(x) are the basis functions that we are projecting onto. So g_k = e^ikx and the complex conjugate is the e^(-ikx). The integral is determining the coefficients (this is the same as doing a change of basis in linear algebra but with continuous functions)

[zachzach]

Great video! There is a minor issue @11:00: c_k should be 1/(2pi)*<f, psi> instead of <f, psi>

[individuoenigmatico]

To be precise the proof that if f(x) is real then for all n, C_n=Č_-n depends on the fact that f(x) is real iff it is equal to its complex conjugate. So you just need to take the complex conjugate of the Fourier expansion of f(x) and equal it to the Fourier expansion itself. Since the functions e^inx are linearly independent one obtains C_n=Č_-n for all n.

[ISKportal]

Is it me or y'all find This playlist super exciting.

[garekbushnell]

Are these infinite orthogonal functions "pointing" in different directions in frequency space? And is there a comparable way of describing the directions in physical space in terms of a single function, the way e^ikx does?

[guest_of_randomness]

it's a good view... thx for it. but how can i proof that there isn't any direction got missed out to form any function?. how can i know that all the direction i need to make a function is inside ψk of -∞ to ∞?

[kin_]

<f(x), g(x)>, , , , so the f is the function we are trying to approximate, what the heck is the g(X) ? it doesnt seem to be defined anywhere ( other than graph few vidoes back maybe ) still not clear of its purpose

[nishapawar]

i feeingl like i wann learn everything just from you, ....lol

[goodlack]

I am sorry, can anyone point me to the explanation of why we are summing from k=-inf? It seems like a bottleneck for me now, the rest and everything that follows I understand.

[yeorinimsida]

I've lost my way from 09:56 ...😭😭😭😭😭

[mehershrishtinigam]

so that's how you're supposed to say L'Hopital ...

[prettycillium]

well, your impact on society after taking the decision of sharing videos about your knowledge is an important asset. Thank you very much for your brief and clear explanation of Fourier analysis. I have watched all videos from this playlist and founded decent as an electronics engineer from Turkey. Thank you again, Mr. Brunton

[zacharythatcher]

This has been really helpful for my understanding of quantum mechanics. Professors always pretend to intuit that the easiest solutions to the hamiltonian are the complex periodic basis functions. Then when they say that they form a complete and orthogonal set of basis functions for all solutions I have a really hard time believing it. Breaking down that the Fourier series was invented as an eigenfunction/Hilbert space solution to PDEs on purpose really helps me digest that the basis is complete by grounding me in the fact that it is just a Fourier representation of any potential answer. Seeing you go through the proofs that quantum courses always go through, but in a more general sense for complex Fourier representations, really helps me understand that these aren’t magical properties of quantum mechanics that someone came up with by analyzing schrodinger’s equation, but rather intuitive properties of the mathematical tools that we are using to understand quantum phenomena.

[ch]

Explanation and derivation of the concepts are so clear! Thanks for uploading the course :)

[MiguelGarcia-zxqj]

No, sir; by proving that a set of functions are mutually orthogonal in [-pi, pi], you do not prove that they form a basis of the square integrable functions on [-pi, pi]. And the proof is trivial: just drop some of the functions in a true basis (in fact the set of all psi[k] form one, but that's not the point), then you have a set of orthogonal functions unable to express the dropped ones.
What I mean is that orthogonality alone is not enough to form a basis of a Hilbert space; you have to prove completitude (or density of the linear combinations, if you prefer).
Ok, I see that the issue has been commented before. But still my counterexample could be of use ...

[thaigo]

Instead of using limits or series expansions you could just say that the integral is given by that expression when j != k and by int[e^(i*0x)dx]=int(1dx) from -pi to pi when j=k.

[erickappel]

Steve Bruton explains so well. He deserves a statue!!!