Zeta function in terms of Gamma function and Bose integral
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- Опубликовано: 25 июл 2024
- Zeta function and Gamma functions,
Bose Integral,
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Hey!, That's the Bose integral right? (Or something with a similar name)
It comes up in Statistical mechanics quite some times.
Nice!!!
Yes it is!!!
blackpenredpen Wait, did you just change the title of the video or did I just not notice it before?
Gurkirat Singh
I just changed the title. Since you reminded me.
Gurkirat Singh as a thank you. I will also pin your comment now!! Thank you!
blackpenredpen I feel honoured. Thanks
Btw since I am talking to you, One thing I never understood, why is the gamma function defined the way it is, I mean why couldn't they just define it to be the same as the factorial (for positive integers), and that -1 in the integral always seems a bit weird.
I know that there's a PI function which fits my purposes, but Gamma is so much more in use. Why would the early mathematicians (who loved it's elegance) choose to define it as such?
It will probably have some physics or application related reason, but nevertheless, it is worth asking
I feel like there should be a line in a horror movie where the antagonist slowly says "we are in the u world now."
"Prepare to be integrated."
Is this better of worse than to be differentiated?
AndDiracisHisProphet - isn't that the equivalent of being dis-integrated💀
only if you are constant
@@elijahshadbolt7334 OH GOD
t = nu then tt = nut
nut. Haha
Oon Han yup! Hahha
Oon Han when you read it out: titty equals nut
Haha aren't you a bit too young for such jokes
This is not for kids
*isn't it*
This is very useful in the Stefan-Bolzmann's law (black body irradiance) to get the sigma constant
Well, for everything related to blackbody radiation you need this.
For example, the number density of photons is given by this integral too.
incredible application
Also the Sommerfeld expansion
This boi is a really nice one as it appears everywhere in quantum statistics in cases where the fugacity is equal to one.
wow! i've seen integrals like your example near the end in statistical mechanics and elsewhere and never knew how they were evaluated. the textbook just gave us the value.
I'm a physicist and I find this kind of integral quite often. We always say "some mathematician proved this result" but I never actually checked it. Nice to see the proof, good job BPRP
I had a question:
Is it legal to use n first as a variable and then take the sum? It feels very sketchy.
Your best video so far. Truly interesting!
Hi! I love your videos, and your math is on point Keep it up 😉
Sure wish these videos were around when I did my degree!
These are great!
Amazing and beautiful result! I always snapshot these kinds of things bc that are so interesting! Thanks for sharing as always!!!!!!!!!! 😇😇😇😇😇
Great work, really interesting as always!
Finally I'm able to correct you. At 4:24 you say "x to the nth power" :D
But it's still a great video =)
I was about to say that! :-(
You are simply awesome Prof. 🙏
This is like the first or second page of Riemann’s paper on primes.
I think i asked for this integral time ago, while i was studying statistical mechanics, but i only saw the video today! great!
What a great video 😍😍😍
Just like usual 😊
WOW!! This is so cool!
This is equivalent to showing that the Mellin transform of the function 1/(e^x -1) is the product of the Gamma function with the Riemann zeta function
This was a question on my final exam of intro to real analysis (maa 5055) and I used this technique!!! I got stuck on proving the integral at 5:20 is uniformly convergent so I was unable to continue forward after that step:( but I’m glad to know I was doing the right thing lol
I recall seeing this result in my statistical physics course when we dealt with bose-einstein distribution. I never saw how this mathematical derivation, though. :) Awesome vid!!!
This is awesome!!❤
You might also see this expressed as the product of the gamma function and the polylogarithm function Li_s(1) (which is zeta(s)). Polylogs crop up in related things like integral of x over exp(x) - 1
This. Is. Awesome.
Gold as usual
390 likes and 0 dislikes... that’s the most likes I’ve ever seen on a video with 0 dislikes. Keep up the good work! (:
oh... 我只在乎你 nice choice for an ending song ! :) Thanks for the uploading !
Thanks!!
that outro music had a Scott Joplin vibe, v-much appreesh
Wow...these things fascinate me...I don't know much about them...but the time will come I promise!
Hi, I'm from Bangladesh. I very much like your videos. Thanks for making video on Bose integral.
Love it!
We need more videos about this,,, BPRP!!!!!!!!!!!!!
:V
Just amazing...
Man, this is very cool. As a recent ME grad, I've been out of the calc3 maths world for a little while, but this makes me want to jump right back into it.
Dean Congrats! :) also, out of curiosity, how much math were you required to take as an ME major?
Thanks Jeff. After doing my calc series, I also had to take a linear algebra and an ODE differential equations course.
you are very cool and very educational
Very nice!
thank very much you are a genius
I'm not sure if this is true, but I think I've heard from somewhere that Fubini's Theorem (or Tonelli's) only works when the function is continuous in the given interval. But then, I don't think we could integrate something that's not continuous in a given interval (unless we break it up into continuous intervals).
But very interesting video! 😯 I bet the next video is "Proof of Riemann's Hypothesis".
Very good!
this video very good. I want that you share such a this video. I'm wacthing from Turkey.
😘😘ابداع يا استاد .احسنت 👏
and if you replace the zeta function by the alternate eta function, you change the sign in the denominator in the integral, from e^u - 1 to e^u + 1.
beautiful
1 fact-oreo!
Bringing back the fun
Amazing
Wow Bose Integral🤩
Wonderful
That was a great one. How can we solve zeta(3)
You can change the order because functions are non-negative. It doesn't need to converge. If you get infinity in one case, you get it also in the other.
If you read Planck Vorlesungen you find 1+1/2^4+1/3^4...=Zeta(4) for Stefan Boltzmann Integral. You can Take any Zeta of even Numbers of zeta, If you adapt the prefactor.
Awesome video! You can't swap integration and addition with this thing: ∫ ∑ x²(1-x²)ⁿ dx for x∈[-1; 1] when you sum over n c:
But why?
This is fucking amazing
very enjoyable
Thanks
Each and every day you look like Dr Peyam
High level maths!
I wonder what would happen if you did this with the eta function you just made a video on
I think for a counterexample where the exchange of sum and int doesn't work you have to come up with some really stupid functions like f_n= a_n*x from 0 to 1/n with a_n so that f_n(1/n)=1 or something like that
AndDiracisHisProphet Excuse me man
I can see you've got a great understanding of maths and physics
Never thought of making some vids? (I would love to see them)
How do you know I have understanding of math and physics?
Also, I would probably make them in german^^
Also, I am ugly af. You wouldn't want to see me :D
AndDiracisHisProphet Well I've read your previous comments in many other math channels and they don't seem to be written by someone who doesn't understand
No problem if you do them in German
Don't say that you're ugly
I personally don't care about that
When you found my comments on other math channels you probably have already noticed that I make a lot of jokes. Not necessarily funny ones. The "ugly" comment was such a joke.
Anyway, I indeed have planned doing math videos, but not using this user name (this is quite a private one), because I am an actual self employed tutor and I will use my companies name.
Thing is I will mainly do stuff for grades 6 to 12, or so. Probably not so interesting for someone watching this kind of videos.
AndDiracisHisProphet But if one day you decide to do some calc, ODE, Linear álgebra. Abstract algebra vids
You've got a subscriber!
I was getting ready to yell at you for casually swapping sums for integrals, but you passingly referred to absolute convergence. This is one of those I'll take your word for it :-)
After doing some homework of my own on this, it seems like a much safer assumption than I would have thought that you can make that kind of interchange. According to single-variable special cases of the Fubini/Tonelli theorems, if int(sum(|f(n))|) < inf, or sum(int(|f(n)|)), then the two are interchangeable for the entire function, sans absolute values.
Tonelli's hinges on Fn(x) >= 0, and I start to lose the trail after that, but I think you're really quite safe. So if you found a function who had negative values and a sum which is conditionally convergent and married the two... maybe??
Also, good job.
Also also, can we get a LaTeX editor built into RUclips Comments?
Bernhard Riemann!
Now, evaluate the integral using the residue theorem, thereby deriving the reflection formula for zeta!
perfect
You should record in 1080p 60fps. Quality res videos for quality maths 👌🏼
TheOnePath he records on mac
60fps sucks
10:35 Bravo!!!
proud to be a bose
Does anybody know the feeling of Depression i have it at the Ende of the Video it is so good why it has to be so good thanks for showing
Thanks.
Are you going off HM Edwards' "Riemann's Zeta Function"? I'm reading it now and this is in one of the first few chapters.
2:06 apparently you sound exactly like me according to my phone. You saying "and take a look" in the video triggered my "Ok Google" command and searched for "look". :l
Small speaking mistake at 4:24, but still an awesome and creative video!
👌👌👌 Well done 🥀🌷 Bro👏👏👏
U just solved the Riemann hypothesis
Let the equation below accept a single solution(n) specify both(a,b) in terms of (n)
f(x)=X^2-(a+b+1)X+(ab)=0
since f(x)=0 is equivalent to
x= (x-a)(x-b)
I think in this space there are zeros of the zeta function .
Great
Hey, I know how to get the result pi^2/6 using fourier series for function x^2. In my opinion, it is the easiest way, could u share yours?
In complex numbers, if "s" is complex, e^(ns) != (e^s)^n... be aware of it because complex exponentiation is not a univocate value operation... so surely the relation is true for real "s", but for complex "s" you could find issues related to the multyvalued results of complex exponentiation
what if you have gamma(x)*zeta(y)
or even maybe gamma(x)*zeta(2x) or something like that.
Can you do anything?
cool~
thats cool...
Please solve integration root Sin(x) dx by udv ?
Blackpenredpen, I am pretty sure that you can always switch the order of summation and integration
zeta(3) = 1/2 integrate 0 to infinity u^2/(e^u-1) by u
Archik4 a close approximation is pi^3/26
refreshing
I'm early and dropped my like
Can we not do something similar with the eta function
What if it's exp(X)+1 instead of -1?
Hey nice video! But I just have one question; when saying a geometric series has the sum of a/(1-r), doesn't the sum need to start at 0 and not 1 like it did in the video?
Suchetan Dontha it depends. That's why I put down "first"/(1-r)
Hold on, x is always a variable, I think it should primarily be converted into something with u.. unless u r doin partial integration, which doean't seem to be the case, is it?
4:11 How were you able to take the gamma function out since it contains n terms?
It doesn't contain any n.
I do not really understand when and why you are allowed to change summation and integration. Can anyone explain is to me?
ζeta if the integral is absolutely convergent (the integral of the absolute value is finite)
Thank you, but I was rather talking about why you're not allowed to change if the sum is divergent
ζeta if a function can be expressed as an infinite series and if it has a radius convergence, and if f(x) is differentiable and can be integrated.... then you are allowed to perform calculus on the series which means treating terms like constants and variables of integration. Meaning you can interchange the summation and integration. However you will have to find a proof of this theorem :)
Because the integral and sum operators are linear and are operating on distinct variables.
ζeta www.maths.manchester.ac.uk/~mdc/old/341/not7.pdf
Just thinking how he hold 2 pens together and switch them while writing.....
Bose condensate. is winter coming ? :)
4:17 I’m confused, if T(x) is defined in terms of n how can you pull it out of the summation?
Can you find ∫0→∞(-cosx/x^sinx)*x^e^x^cscx? I tried to find it using an integral calculator, but it gives me no answer. :(
I forgot, where do you teach again?
To permutate the infinite sum and the proper integral, shouldn't the sum converge uniformely and not only absolutely??
Wow, an amazing proof. The ancient Greeks would be surprised to see Gamma, Zeta, Pi in one equation.
Is there any application of the beta and gamma functions in physics?
Quantum Mechanics are plenty used.
4:25 x to the nth power
Couldn’t you rearrange this to come up with a continuation of the zeta function?
Does this formula work on the critical strip? Trying to graph |zeta(s)| as a function of x + iy (3D plot) did not seem to work. Maybe it's the program I'm using. Also, when I put in the first non-trivial zero of the zeta function, this formula you have does not return zero, so I think maybe it does not work on the critical strip... sadly. The function Riemann gives defines the zeta function in terms of the zeta function of 1-s. And when a function is defined using itself (sort of)... that's where the issues arise. My opinion. I wish there were a better form of the zeta function that did not do that.