A Curious Derivative of an Infinite Sine Power Tower Tetration... - Papa's Improvised Session #10

Today we differentiate the infinite tetration of sin(x) wrt x :) Enjoy! =D
This video has been sponsored by Wondershare btw :)

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[Rundas]

Hagoromo chalk is actually 3 things:
-best chalk
-best waifu
-best cocaine alternative if you also wanna become smart in the process 😂

[alinoon]

6:10 no need to censor nothing my man, we arabs love this channel, all love bro

[kiiometric]

For a moment I realized I was watching a german mathematician speaking english and japanese while solving a problem talking about waifus xD

[arcannite]

it's actually infinite penetration

[musa]

"out of my good mythical morning cup... back when it was a good show" ;-;

[joseftrojan]

Bruh, I wish I could be a god mathematician memelord like papa.

[PrincessEev]

I'm not entirely convinced of the convergence/well-defined-ness of the infinite tetration. The infinite tetration of x for example only converges for x in [1/e^e, e^1/e] which is about [0.066, 1.44], i.e. an interval in the positive reals. However, sin(x) not only is sometimes outside of this interval, but sometimes negative, and negative numbers to other negative exponents (save a few special cases like integers) behave very badly. You can't even get a continuous real graph of f(x) = sin(x)^sin(x) for instance.


Granted this is solved by simply restricting the domain of sine as deemed appropriate.

[angelmendez-rivera]

I know some of the comments have already addressed different parts of the video, but I want to write a unified comment addressing everything myself all in one comment, and also because I want to correct a lot of the falsehoods stated in some of the comments.

3:17 The power tower x^x^x^••• has standard Cauchy convergence when x satisfies the inequality 1/e^e < |x| < e^(1/e), where x is complex, and this inequality has only countably many exceptions. The reason I am specifying that x can be complex and not just positive real is because it is a well-known fact that the power tower i^i^i^••• converges, and this constant even has its own name, though I forgot. Seeing that i is the imaginary unit, it obviously is not a positive real number, but it still satisfies the above inequality. As such, the power tower sin(x)^sin(x)^••• converges if 1/e^e < |sin(x)| < e^(1/e). Every complex number x that satisfies this inequality is part of the domain, and I believe the power tower is in fact complex-differentiable in this domain, since I believe this domain is an annulus.

Also, many people below have said that there is no value of x such that sin(x) < 0 and the tower still converges. This is false, and it is false because of the inequality above. It is true that the result is not a real number, but this is irrelevant. Functions with a complex range can still be real differentiable, although in this case, I believe there are infinitely many regions with this condition where the tower is not real differentiable or even complex differentiable.

You can differentiate the tower directly without using implicit differentiation. You can find a closed-form for the tower by solving the equation y = sin(x)^y for y. This can be done by using logarithms and Lambert functions. log(y) = y·log[sin(x)] ==> y^(-1)·log(y) = log[sin(x)] ==> log(y)·e^[-log(y)] = log[sin(x)] ==> [-log(y)]·e^[-log(y)] = -log[sin(x)] ==> -log(y) = W(-log[sin(x)]) ==> y = e^[-W(-log[sin(x)])] = -W(-log[sin(x)])/log[sin(x)], and the fun fact is that this satisfies the differential equation that you incidentally derived when doing implicit differentiation. So, of course, you can differentiate this, and you can even do this without worrying about convergence if you consider another type of convergence that is not Cauchy in nature.

Finally, it is perfectly fine to acknowledge 0^0 = 1. The tower still would not converge for x = 0, though, because 0^0 = 1 implies 0^0^0 = 0^1 = 0, and 0^0^0^0 = 0^0 = 1. In general, you can use induction to prove that 0^^n = 1 if n is even and non-negative, and 0^^n = 0 if n is odd and positive. This implies lim 0^^n (n —> ♾) does not exist.

[strengthman]

I wonder if you can get a closed form for the sin tower by solving that differential equation

[Knuckles]

If sin(x) > 0 then y -> 1 then dy/dx = 0
if sin(x) < 0 then y does not exists on real axis since we can't calculate things like (-0.5)^(-0.5). It could exists in complex space but still be undefinable and then dy/dx does not exist.

[anshusingh]

Converges for: 1/(e^e) <y<(e)^(e/2) ....oily day today

[RCSmiths]

Dude, another well done and in depth mathematical enlightenment! Always a pleasure to tune in!

[NrSgt]

Drinking coffee to get to a 10min video. That's what I call a pro gamer move

[vivekbooshan]

Dude, the thing you said about Good Mythical Morning is beyond accurate. After Link lifted his hair, it all went downhill from there.

[mihaipitigoi]

As a fellow physicist i can aprove the accuracy of your t shirt! Your videos are really good keep up!

[ttnrg]

Papa you’re the only youtube channel I turn off adblock for because I know it’s a good cause

[TheNiTeMaR]

I could never tell what accent my lecturer had until one lecture he said LOGARITHMUS NATURALIS and immediately all was clear

[nicholasbohlsen]

I plotted this power tower function in python and it turns out is only converges in the reals for regions where sin(x) is positive and non-zero. At points where sin(x) =0 it alternates between 0 and 1 (if you accept 0^0=1) for each finite approximation, and since a negative number to a negative power isn't well defined in the reals, it cannot converge for the negative sections of sin(x)'s oscillation.

[dackid]

Despite the meme, you are one hell of mathematician Papa Flammy. :)

[Gameboygenius]

I wonder how much the final answer can be simplified. I can see one trivial thing you can do. y*log(sin(x)) can be written as log(y) by moving y to the exponent inside the log.