a nice product from Ramanujan -- featuring 3 important constants!

[@endersteph]

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

[@robertveith6383]

arctan(z)/z dz. Don't forget the grouping symbols.

[@wolfmanjacksaid]

Did he really do it by mistake? He wrote it again at 17:47

[@martindedek2885]

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

[@MichaelRothwell1]

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. ;)

[@idjles]

Yeah, and Ramanujan did that while you were making your coffee for breakfast.

[@mohammedsenhaji]

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.

[@giannipiccioni8411]

​@@mohammedsenhaji it would be only one line with a long enough page

[@乔峯]

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.

[@humbledb4jesus]

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...

[@Noam_.Menashe]

Where do you see number theory here?

[@polyhistorphilomath]

@@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.

[@daburak2559]

You watched the video and you thought of number theory?? How

[@macicoinc9363]

@@daburak2559euler’s prime product isn’t number theory? What is number theory to you if that isn’t apart of it?

[@robertveith6383]

​* a part "Apart" means "separated by a distance."

[@slavinojunepri7648]

What an awesome proof! Amazing to say the least.

[@ഡിങ്കനേഷ്]

That's awesome - from "the man who knew infinity "

[@angeloluisrocattojunior3425]

I'm wondering what else Ramanujan would produce and discover if he had lived more than 32 years.

[@張謙-n3l]

He's probably just a messenger

[@ccg8803]

Your videos are amazing, thank you! I cannot understand the views/quality ratio!

[@franksaved3893]

Why is all the stuff about Ramanujan always fckin' crazy? 😅

[@sanjogar]

Ramanujan was an alien 😂👽

[@vasilisr7]

Thank you Michael

[@curtiswfranks]

This one was pretty tricky.

[@darkseid856]

He predicted 4G technology . Truly ahead of his time .

[@sahilsingh6048]

Tumnay profiel picture change karli kya

[@om5621]

...what? he had nothing to do with even the internet whatsoever, is this some kind of joke

[@Cccctyt]

​@@om5621also black holes

[@aug3842]

@@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

[@cristiandalessandro599]

​@@om5621 yes, that's pretty much the best, most unpredictable joke. Quite hard to not understand it

[@gp-ht7ug]

I miss the initial music of the videos. Very interesting video!

[@한매쓰]

It was a very fascinating content

[@minwithoutintroduction]

رحلة شاقة لكنها ممتعة

[@missoss]

I love these random proof videos.

[@d.h.y]

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 😱😱

[@natepolidoro4565]

Doesn't adding that n=0 term make the whole thing equal to zero as x->1?

[@danmcalister2215]

And then taking the log -> infinity?

[@mohammedsenhaji]

Yeah but he doesn't take the limit until substracting the term again.

[@curtiswfranks]

I think that this is the trickiest part.

[@idjles]

I am sure the Ramanujan saw this identity immediately- he didn’t need 20 minutes of derivation to prove this.

[@TheEternalVortex42]

Ok well since you're "sure" I guess we know it for a fact

this is a very common misbelief

[@damascus21]

Ramanujan was a genius; he wasn't magic.

[@idjles]

@@damascus21 he is so far beyond us, it may as well be magic.

[@yuseifudo6075]

​Nah @@idjles

[@reeeeeplease1178]

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?

[@adogonasidecar1262]

What happened at 14:21 when 4/Pi became Pi/4?

[@adogonasidecar1262]

Oh he corrected it at 15:00

[@Maths_3.1415]

Interesting...

[@Velociraptor103]

Quite cool

[@euler73]

Is "Oh look, we've got...." then new "we can play the same game again..." ? ;-)

[@emanuellandeholm5657]

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?

[@Hexer1985]

Okay, great. So, the three important constants are e, G and \pi.

[@jawadibrahim2367]

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.

[@evid-rz3nu]

nice video sir ji

[@taranmellacheruvu2504]

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?

[@mohammedsenhaji]

Are you talking about the left side ?

[@taranmellacheruvu2504]

@@mohammedsenhaji Yes. My bad

[@insaanh00n]

Clearly Ramanujan was crazy or inspired. Who else would come up with that formula?

[@evid-rz3nu]

love from india

[@suzune8952]

Its pretty normal deriving this stuff , it isnt an alien thing...

[@erfanmohagheghian707]

I did the derivation of 1/cos(pi*x/2) on your behalf :) Thank you sir kzbin.info/www/bejne/rHKTgmdjf8yXhLc

[@Packerfan130]

so many cheats, why bother doing the video at that point then?

[@EphemeralEphah]

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.

[@zlodevil426]

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?

[@TheEternalVortex42]

All math is built on earlier work. It's fairly arbitrary which results we would want proven and which are considered known.

[@cameronbigley7483]

​@@zlodevil426I don't mind referencing other videos, but I'd ptefer if an important step was already uploaded prior.

[@samuelgoodman6067]

@@EphemeralEphah It's actually one line. Use the Taylor series for arctan.