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In physics, conventional superconductors described by BCS theory also show this exact beautiful fraction of (pi/e^gamma). The ratio between their gap function and critical temperature for the transition gives you this number.
It is an ‘approximate’ formula (which I find even more intriguing) at zero temperature. Also interesting is that the ratio near critical temperature becomes of the order pi/sqrt(zeta(3)), which if you for once forget the derivation for, feels almost supernatural!
Slight error. One may switch the order of summation of two infinite series if each series is ABSOLUTELY convergent. Mere convergence is insufficient. Also capital "S" looks almost exactly like Zeta.
Better yet; A transcendental node with a dimensional orientation and relative aspect ordinal space formation suitable for blueprinting serial numerical architecture.
Even Better; Standing wave of the product of the zeta primes? Just a guess by exclusion of the remaining rational possibilities, and, an excape from humiliating exaction on the re-ordered spindle of confined charme.
Zeta is the sum of reciprocal powers, for example zeta(3)= 1 + 1/8 + 1/27 + 1/64 + … S includes the sum of all integer zeta functions, zeta(2)+zeta(3)+zeta(4)+… (of course, each term divided by n 2^n) So S is essentially a double summation, and zeta is a single summation.
It would be easier to see the beauty if there were some more motivation for the lengthy derivation, and some explanation for what insights it gives. Even in its final form it looks very complicated on the left-hand side. The fact that an exponential of an infinite series on the left equals the inverse of an exponential (i.e., another infinite series) on the right hardly seems remarkable without more context.
Yeah, there too much, explore mathematic constant, and some book of encyclopedias of Series and Integral like … there to many, what i surprise why there are no beautiful mathematical and physical book like old time. Bull…shit internet media.
@@maths_505i have a question here is it possible to analytically continue the function beyond (-1,1) using your result here with the log of the gamma function which is of course well defined on the complex numbers?
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If you like the videos and would like to support the channel:
www.patreon.com/Maths505
You can follow me on Instagram for write ups that come in handy for my videos:
instagram.com/maths.505?igshid=MzRlODBiNWFlZA==
In physics, conventional superconductors described by BCS theory also show this exact beautiful fraction of (pi/e^gamma). The ratio between their gap function and critical temperature for the transition gives you this number.
Better Call Saul theory
It is an ‘approximate’ formula (which I find even more intriguing) at zero temperature. Also interesting is that the ratio near critical temperature becomes of the order pi/sqrt(zeta(3)), which if you for once forget the derivation for, feels almost supernatural!
this channel is so chill and good. love your work boss
Thanks
This is indeed extremely beautiful!
It would be, if I could read it. The handwriting sucks.
As a generating function enjoyer I SIMPLY CANNOT HELP BUT NOTICE that this looks like a derivative of zeta gf boi at 1/2
LOVELY RESULT
Awesome Proof. Thank you for your innovative video.
Very beautiful.
The integral of fractional part of tanx from 0 to π/2 has gotta be my fave though.
I remember integrating {tanx}/tanx but just the {tanx} sounds like a good idea.
Noted. Thanks mate.
I remember the result looks quite terrifying, right?
That was absolutely awesome!
It is beautiful to see e gamma 2 and tau in one place
Slight error. One may switch the order of summation of two infinite series if each series is ABSOLUTELY convergent. Mere convergence is insufficient.
Also capital "S" looks almost exactly like Zeta.
With the restriction |x|
Bro it is indeed a pure gold. Where did you find this?
By accident
F*CKING BEAUTIFUL
Been a while for me, but can't you rewrite exp ^ series to be an infinite product?
Yes ofcourse and that would be pretty cool too
Very nice
Very satisfying indeed!
Convergent is not enough to interchange the order of summation. You need absolute convergence.
Lovely!
Nice video, in which app are you writing?
What software is this with the black screen and the icons at the bottom?
Samsung notes
A transcendental node and dimensional orientation relative aspect sign?
Better yet;
A transcendental node with a dimensional orientation and relative aspect ordinal space formation suitable for blueprinting serial numerical architecture.
Even Better; Standing wave of the product of the zeta primes?
Just a guess by exclusion of the remaining rational possibilities, and, an excape from humiliating exaction on the re-ordered spindle of confined charme.
8:08 can someone tell me the difference between zeta and s
Zeta is the sum of reciprocal powers, for example zeta(3)= 1 + 1/8 + 1/27 + 1/64 + …
S includes the sum of all integer zeta functions, zeta(2)+zeta(3)+zeta(4)+… (of course, each term divided by n 2^n)
So S is essentially a double summation, and zeta is a single summation.
Bernoulli boyz
It would be easier to see the beauty if there were some more motivation for the lengthy derivation, and some explanation for what insights it gives. Even in its final form it looks very complicated on the left-hand side. The fact that an exponential of an infinite series on the left equals the inverse of an exponential (i.e., another infinite series) on the right hardly seems remarkable without more context.
7:40 - You forgot the (-1)^n in the sum.
Cool
Noice :)
Amazing result thank you Sir❤❤❤
A+++++++++++
Yeah, there too much, explore mathematic constant, and some book of encyclopedias of Series and Integral like … there to many, what i surprise why there are no beautiful mathematical and physical book like old time. Bull…shit internet media.
schwifty
Indeed
Let x=1
On the way to deriving the result, I specifically mentioned that abs(x)
@@maths_505yeah I forgot then I will try 1/2 this must work?
@@karma_kun9833 yes 1/2 will work perfectly
@@maths_505i have a question here is it possible to analytically continue the function beyond (-1,1) using your result here with the log of the gamma function which is of course well defined on the complex numbers?
@@ΙΗΣΟΥΣΧριστος-θ2γ in deriving the result I used the geometric series so one would have to start from scratch using a different approach.
“Most” word must be prohibited in youtube titles
Why