Deriving EULER's INFINITE SINE PRODUCT using the Mittag-Leffler Pole Expansion of the Cotangent!

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Sry for the sound problems!
Today we are going to derive a very important number theoretical identity! We are not going to take the heuristic approach euler once did, but rather a more rigorous approach to the whole thing. Let's decompose sine into its linear factors, enjoy! =)

[The_Professor_S_]

Mortal Kombat 11’s true final boss: Papa Flammy, The Machine Of Argumentation

[AA-wqzy]

In math forums, the expression is always said to be derived from Weierstrass factorization theorem, but this is a new and different way to prove this, thanks!

[ianmathwiz]

"...but I actually prefer the rigorous approach a bit more."

And with that, Papa Flammy declared war against the physicist camp.

[helloitsme]

From this identity you can also easily see that the limit as x->0 of sinx / x is 1 when you divide both sides by x. And yes that's allowed cause you're taking the limit

[danielcastillo]

What a perfect segue into fractional calculus!!!

[frozenmoon]

The infinite boys laughing at the beginning really got me. You know papa flammy gonna make some infinite joke for ye, before the actual fun begins.

[rabindranathghosh]

Wow this identity is so useful in showing that sin(x)≈x for small values of x

[gregoryfenn]

'whale... cumback'

I was not expecting this kind of joke on a maths lecture :O

[MooshPeriwinkle]

Hi Flammy!

Got a little question at time stamp 7:36, what functional equation are you talking about in complex numbers that allows us to make the switch from series to product? Doing a lil complex analysis project and this would be so useful.

[wuyqrbt]

Dear Papa ...
At that time you were sick but you did your best as always.
Great video with interesting results (combine it with previous video, cot).
Thank you so much 💖

[oswinheinrich]

Proposal for discussion: what do you get when you replace the minus by a plus in the infinite product. Isn't it sinh(x), but how goes the proof?

[maxteer]

swear i'm not the only one who feels this, but you are my spirit animal

[Exilelord]

This was so amusing to watch, you should have a *Maxmoefoe* meltdown in one of your videos

[dectorey]

6:20 TOP 10 MOMENTS RIGHT BEFORE DISASTER

[Davquest]

Microphone: *rubs agains shirt*
Literally everyone: NANI?!

[hungryfareasternslav]

Engineers: sin x = x
Newton: *Throws trillions of apples at engineers*
Leibniz: Newton, Isaac, you are noob. *Cut all engineers with his integral sword*

Newton: What!!!

[josephholten]

Papa, as always great vid, just try to keep your mic from rustling :)

That was damn exciting ! Btw ... that chalk has no right to exist on this planet any more ... Hmm, is it a spoiler to have a video from another planet next time? :-O Good vacation plan btw ... another planet.

[akshat]

your eyes look really messed up. I know this was recorded a while ago but are you okay daddy? you don't have to stress yourself out so much to put out videos

[swordslash]

I dunno why this was recommended to me but I enjoyed it