At 15:14 he forgot to change the inside of the integral to arctan z / z dz, but it doesn't change the result because the integral goes to 0 in the limit because of the bounds
I think he didn't want to substitute it to z, just the other two integrals. So I think that the mistake was that he changed the higher bound to tan(pi/4...), I think it should've been only pi/4... But still for X=1 it's 0 so the integral is irrelevant
number theory is like getting a nasty sliver...you hate it, but you can't stop looking at it, picking at it, and finally getting knife to try and dig it out...
Piece of cake. All he did was add a variable x, take logs (to convert infinite product to infinite sum), differentiate, simplify, integrate again, use a bunch of substitutions and a series representation of sec, and a (familiar) integral representation of Catalan's constant. ;)
And don't forget that you have to check the conditions for every use of limit, derivation, integration rules for series and integrals involved. If written rigourously it is at least 6 pages.
New theory or formula may seems easy becos the inventor or discoverer tried many ways to simplify it as direct n simple as possible before presented it to you. But before that, nobody had the correct and simplest answer. They have tried many old, new, harder ways in order to find the correct one that is error proof. And this isn't really that easy or obvious, otherwise it would have been discovered way before he/she did, it is the unique creativity and ingenuity that lead each scientist made the contribution of their own. The sea of knowledge is so vast, no single person is smart or creative enough to discover n understand everything in the universe. We should respect and give the credit to all the scientist for all the discovery n invention they made, even if it is only 1 small equation, cos every equation count.
@@om5621in the exponential there is the term 4G/π, and if u ignore the π in the denominator and treat G as just a letter rather than Catalans constant then it reads ‘4G’, which Ramanujan found well before the internet was created
Imagine you are a professor and all of a sudden a student brings up this identity in one random morning, while he brings up some incomprehensible pi formula (1/π =...) in the other day 😱😱
I was wondering for a solid minute why Newton's gravitational constant popped up in a mathematical result, only for it to be a different G in this case. I'm a little shook right now.
I don't know about you guys, but my little ape brain was screaming difference of squares factorization as soon as I saw factors of the form [ 1 - ( 1 / (2n + 1) )^2 ] to the f(n). Is that something to explore?
Would you want the video to be well over an hour long? I assume proving the integral representation of G is equal to the series definition would take quite a while and could be it's own video, and I imagine deriving the series representation of sec((pi/2)x) is not so simple either.
A lot of his videos reference other videos, most reference ones in the past, and this one just references videos in the future. What’s the significant difference?
At 15:14 he forgot to change the inside of the integral to arctan z / z dz, but it doesn't change the result because the integral goes to 0 in the limit because of the bounds
arctan(z)/z dz. Don't forget the grouping symbols.
Did he really do it by mistake? He wrote it again at 17:47
I think he didn't want to substitute it to z, just the other two integrals. So I think that the mistake was that he changed the higher bound to tan(pi/4...), I think it should've been only pi/4... But still for X=1 it's 0 so the integral is irrelevant
number theory is like getting a nasty sliver...you hate it, but you can't stop looking at it, picking at it, and finally getting knife to try and dig it out...
Where do you see number theory here?
@@Noam_.Menashe There's the product over factors of the form (1/(1-1/p))^c_k . That's probably a good indication number theory is involved.
You watched the video and you thought of number theory?? How
@@daburak2559euler’s prime product isn’t number theory? What is number theory to you if that isn’t apart of it?
* a part "Apart" means "separated by a distance."
Piece of cake. All he did was add a variable x, take logs (to convert infinite product to infinite sum), differentiate, simplify, integrate again, use a bunch of substitutions and a series representation of sec, and a (familiar) integral representation of Catalan's constant. ;)
Yeah, and Ramanujan did that while you were making your coffee for breakfast.
And don't forget that you have to check the conditions for every use of limit, derivation, integration rules for series and integrals involved. If written rigourously it is at least 6 pages.
@@mohammedsenhaji it would be only one line with a long enough page
New theory or formula may seems easy becos the inventor or discoverer tried many ways to simplify it as direct n simple as possible before presented it to you. But before that, nobody had the correct and simplest answer. They have tried many old, new, harder ways in order to find the correct one that is error proof. And this isn't really that easy or obvious, otherwise it would have been discovered way before he/she did, it is the unique creativity and ingenuity that lead each scientist made the contribution of their own. The sea of knowledge is so vast, no single person is smart or creative enough to discover n understand everything in the universe. We should respect and give the credit to all the scientist for all the discovery n invention they made, even if it is only 1 small equation, cos every equation count.
What an awesome proof! Amazing to say the least.
Ramanujan was an alien 😂👽
He predicted 4G technology . Truly ahead of his time .
Tumnay profiel picture change karli kya
...what? he had nothing to do with even the internet whatsoever, is this some kind of joke
@@om5621also black holes
@@om5621in the exponential there is the term 4G/π, and if u ignore the π in the denominator and treat G as just a letter rather than Catalans constant then it reads ‘4G’, which Ramanujan found well before the internet was created
@@om5621 yes, that's pretty much the best, most unpredictable joke. Quite hard to not understand it
Why is all the stuff about Ramanujan always fckin' crazy? 😅
Thank you Michael
This one was pretty tricky.
6:51 why don't we just view it (as he pointed out) as -pi/2 x sec(pi/2 x)? We should be able to integrate usec(u) with IBP, no?
I am sure the Ramanujan saw this identity immediately- he didn’t need 20 minutes of derivation to prove this.
Ok well since you're "sure" I guess we know it for a fact
this is a very common misbelief
Ramanujan was a genius; he wasn't magic.
@@damascus21 he is so far beyond us, it may as well be magic.
Nah @@idjles
Imagine you are a professor and all of a sudden a student brings up this identity in one random morning, while he brings up some incomprehensible pi formula (1/π =...) in the other day 😱😱
What happened at 14:21 when 4/Pi became Pi/4?
Oh he corrected it at 15:00
Your videos are amazing, thank you! I cannot understand the views/quality ratio!
رحلة شاقة لكنها ممتعة
I was wondering for a solid minute why Newton's gravitational constant popped up in a mathematical result, only for it to be a different G in this case.
I'm a little shook right now.
It was a very fascinating content
I miss the initial music of the videos.
Very interesting video!
Doesn't adding that n=0 term make the whole thing equal to zero as x->1?
And then taking the log -> infinity?
Yeah but he doesn't take the limit until substracting the term again.
I think that this is the trickiest part.
I love these random proof videos.
Quite cool
Okay, great. So, the three important constants are e, G and \pi.
Is "Oh look, we've got...." then new "we can play the same game again..." ? ;-)
I don't know about you guys, but my little ape brain was screaming difference of squares factorization as soon as I saw factors of the form [ 1 - ( 1 / (2n + 1) )^2 ] to the f(n). Is that something to explore?
Interesting...
nice video sir ji
Its pretty normal deriving this stuff , it isnt an alien thing...
Clearly Ramanujan was crazy or inspired. Who else would come up with that formula?
How can you justify that the limit means an equality? Can’t the point at x=1 be different from the limit as x goes to 1?
Are you talking about the left side ?
@@mohammedsenhaji Yes. My bad
love from india
I did the derivation of 1/cos(pi*x/2) on your behalf :) Thank you sir
th-cam.com/video/vA0Q4-JgdNU/w-d-xo.html
so many cheats, why bother doing the video at that point then?
Would you want the video to be well over an hour long? I assume proving the integral representation of G is equal to the series definition would take quite a while and could be it's own video, and I imagine deriving the series representation of sec((pi/2)x) is not so simple either.
A lot of his videos reference other videos, most reference ones in the past, and this one just references videos in the future. What’s the significant difference?
All math is built on earlier work. It's fairly arbitrary which results we would want proven and which are considered known.
@@zlodevil426I don't mind referencing other videos, but I'd ptefer if an important step was already uploaded prior.
@@EphemeralEphah It's actually one line. Use the Taylor series for arctan.