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

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.

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

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

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."

He's probably just a messenger

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

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.

Oh he corrected it at 15:00

Are you talking about the left side ?

@@mohammedsenhaji Yes. My bad

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.