Fourier sine series of f(x) =x²

In this video, I showed how to find the fourier sine series of f(x) = x^2

[herodotomello]

Man, you até great! I If you lived in Ancient Egypt or Ancient Greece you would be called a hierophant! A hierophant was one who explained Sacred things. Thank you for sharing your knowledge in such a thoughtful and accessible way. Greetings from Brazil.

[johnnolen]

Hey Newton, very clear explanation as usual. 👍 There are a couple of details that you did not mention however. First, Fourier series express periodic functions of the independent variable. [If f(x) is non-periodic then you typically calculate the Fourier Transform. The same basic calculation but without the factor of 2/l out front.] Second, the reason the Fourier coefficients, Bn in this case, are calculated using the same trig function they are paired with (sin [n πx/l]) is due to the orthogonality of trig functions; i.e. Integral[ sin[n πx/l]·sin[m π x/l] dx] = 0 when m != n. Third, the only reason to calculate the Fourier sine series alone is if the Fourier cosine series vanishes. Calculating both the Fourier sine series and the Fourier cosine series results in a complete representation of f(x).
Finally, the reason for doing any of this is to determine how the energy of a signal represented by f(x) is distributed across the frequency spectrum.

[jadenredd]

great video!! i didn’t realize i was gonna enjoy fourier analysis like this until you broke it down 👍🏾

[happyhippo]

Over 30 years ago, I took a whole university class on just Fourier series, during my MS-Systems engineering curriculum. I haven't really used it since but I enjoyed the flashback. I got an A in that class.😀

[TheFrewah]

Hi PM. I think you haven’t done videos on how to calculate roots by hand, i.e. using only a basic calculator. At age 14, I came up with a way which I think is very straightforward. My Aha moment came when I realised that x/√x = √x. That is beautiful, isn’t it. The square root can be thought of as an equilibrium. You divide a number by something and what you get is what you divided by. Take 9 for example. If you divide by something smaller than three, you get something larger than three and the correct value is somewhere in between in the general case. Start with, say, two. 9/2 = 4.5, then calculate the average which is (2+4.5)/2 = 3.25. Now, divide nine by this value 9/3.25 ≈ 2.769. The average is now
3.25+ 2.769)/2 ≈ 3.0095. Pretty decent, don’t you think. The idea can be extended for any integer root. Say you want the cube root of 25. Three is a good starter. The idea is now to divide by the square of your initial guess, so (25/(3*3) ≈ 2.77778, the average is (3+2.77778)/2 ≈ 2.88889. It works with non integer roots as well as long as the reciprocal of the decimal part is an integer.

What’s funny is that I have never seen this method used. I’m really proud to have figured out myself at such a young age.

[renesperb]

Fourier series are very important in the context of PDE's . Hence, boundary conditions have to be satisfied, and this tells you then which sine or cosine series you have to use. The case you discuss here then corresponds to zero - boundary data . I think one should point this out.
One could ask : why do you choose a sine series ?

[hvok]

Phenomenal presentation, always enjoy your videos.

[LenzTrack]

Thank you for this well done explanation.

[glorrin]

French guy here: pronounciation is good :)

[Ron_DeForest]

Wondering if you can talk about the concept of e. Where did it come from, what was the need of it, how it’s derived. Anything else that comes to mind.

[MarcoMate]

When you plug in x = 1 in the final formula, you get the wrong result 1 = 0.

[ThOneWhoWaits]

7:16 Interesting. Whats the name of the rule you referenced?

[diptanshumazumdar]

Sir Can u please make a new playlist on making Graph of every Functions please

[ThOneWhoWaits]

Sir, if you graph the fourier series on desmos with the final answer, it doesn't match. How is this possible?

[sumanbarmanmusic]

Where are you from sir?
Love from India ❤

[alipourzand]

Neat! However I'm not sure that using the Fourier serie of x^2 would make the life easier!☺

[petersprague]

I must be missing something. How is this a simplification?

[rampp]

Could you Make a Video about pi tetrated to pi? Would Love to See it

[ronaldjensen]

5:41 should have ended the integral with a dx or some other d? A lot of variables floating around here...

[andrejflieger]

So in french the last letter is usually not spoken, ou almost sound like oo in fool or root in english. So probably pronounce it like Foorie