when 'normal' trig functions aren't enough -- the Clausen function.

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[emanuellandeholm]

The audio dsp nerd in me instantly recognized this as a phase distorted version of the triangle function. If you "listen" to this function, looped over any interval covering 2 pi, you will essentially hear the triangle function (because phase offsets don't matter that much to our brains), which is a very popular oscillator choice in synthesizers. This is all because the triangle function, and all it's siblings whose harmonics are shifted randomly, has a spectrum that falls off as 1/f^2, which sounds quite mellow.

Incidentally, the sawtooth (another favorite oscillator in subtractive synthesis) has a spectrum that decays as 1/f. This is a much harsher sound that begs to be filtered, which is usually what happens in subtractive synthesis.

Also, the development of that geometric series was the climax for my little brain. I was craving a GS from the get go. :D

Edit: So it's not a pure triangle. A pure triangle only has the odd harmonics, but it will have a similar timbre.

[mattikemppinen]

"some of you might wonder why i put those double angle formulas on the board, well that's because i'm about to use them" :D pure gold!

[bjornfeuerbacher]

3:20 to 7:20: Much shorter would have been to write 1 - e^-ix = e^-ix/2 (e^ix/2 - e^-ix/2) = e^-ix/2 times 2i sin(x/2).

[assassin]

I think its worth noting that the last identity holds if n is a natural number, but can be extended to all integers except 0 if instead of 1/n^2 we do 1/(n|n|)

[yoav]

Nice! For the integral identity we can also prove it by using the series of ln(2sin0.5x) = sum from m=1 to inf of -cos(mx)/m then the integral become to the sum2/(nmpi) integral cos(nx)cos(mx) from 0 to pi this integral is 0 if m !=n and pi/2 if m=n so we get at the end 1/n^2.

[GiornoYoshikage]

I guess the Fourier series representation of 'Cl_2(x)' is true only for 'x ∈ [0; 2π]'. It's derived from Fourier series for 'ln|2sin(x/2)|' which is true on the very interval. The Fourier series for 'Cl_2(x)' is 2π-periodic but intergral represetation of 'Cl_2(x)' implies that the function decreases and can't be 2π-periodic

[abhinavanand]

14:44 cl2 is odd function and sin(nx) is odd function therefore the product is even

[aweebthatlovesmath]

1:30 there are second uncountable infinity many function that can't be defined by series there are second uncountable many functions from C to C and one can prove that functions that can be defined as power series are first uncountable infinity so there are second uncountable many functions that cannot be represented as power series so we don't know most of functions which is blowing my mind!!

[JCCyC]

That function can also be generated by putting two carbon rods in a glass of salt water and connecting each pole of a battery to one of them.

[Happy_Abe]

@14:35 why is it even?
Sin is an odd function so why is this allowed?

[barryzeeberg]

What is the practical use of Clausen functions?

[artichaug]

Thanks Michael, that felt like a heavy lifting exercise.

[fartoxedm]

The real part would be ln(|2 * sin(x/2)|) not the one without the modulus

[jayhem_klee]

What about the *absolute value* of 2 sin x/2 ? We lost it...

[ArthurvanHudt]

Fun thing is that there’s also a hyperbolic version of the Clausen function!

[gregsarnecki]

Great video as always but why would you use the Clausen function? I see a complex dilog video on the horizon...

[endersteph]

What does it mean to take an integral with complex bounds? How do we know that |u| < 1 when calculating ∫ du/(1 − u) from 0 to exp(−ix)?

[binaryblade]

Dude, all cos sin double angle crod and you could have done it in like 1 like be factoring a complex exponential

[franksaved]

17:00 and that's a good place to stop calculation

[trelligan]

@15:33: Writing Pi while saying Theta is a good trick. And that tshirt tab stuck up on your neck is a bit distracting 😏 #FeedTheAlgorithm