Eulerian Integral of the First Kind - Deriving the BETA FUNCTION! [ The non-trigonometric Version ]

[@TopCuber]

> if you have one apple then you only have one apple

[@pythagorasaurusrex9853]

Only real mathematicians understand the beauty and logic behind this statement

[@thephysicistcuber175]

100% accurate

[@fridgepuff1722]

You're really going to hit us with the n=1 and not expand it some more? That's it chief? One apple come on man give us more

[@EpicHero-qe7zd]

"Just Fubini that shit !" - Papa Flamie, 2019

[@2neutrino]

nice Australian meme at the beginning

[@xD-jm2ie]

Nothing out of the ordinary to me, what are you on about?

[@nanigopalsaha2408]

This made me laugh more than the meme

[@hehebwoai3056]

Could you do a series where you derive/discuss every oiler thing in maths

[@lilyyy411]

oily macaroni

[@silentinferno2382]

@@PapaFlammy69 all for itt!!

[@pythagorasaurusrex9853]

This would be one hell of a loooooooong video. Or you can do it in 10 seasons, each with 30 shows lasting ca. 3 hours each :)

[@reinerwilhelms-tricarico344]

Very well explained. I’ve been always a bit nervous handling double integrals like this - so easy to completely screw up. Next you might want to talk about the Dirichlet probability density. There is a generalisation of the Beta function.

[@owenl3929]

I haven’t watched this until now since I’ve had so much work but I’ve been looking forward to this!!!! Thank you for one of only a few videos on this function on KZbin

[@cbbuntz]

The beta function is fascinating and really useful for taking a shortcut in otherwise very complicated expressions. One cool thing is if you fill a matrix with the results of the reciprocal beta function (with some (1/2+cos(n+k)/2) terms and dive the first column by 2), and then take the inverse of that matrix, you get the chebyshev polynomials of the first kind. Chebyshev polynomials are orthogonal relative to a weight of (1-x^2)^(-1/2), but orthogonal with no weight when applied to cosines (because the cosine transform is already orthogonal). Chebyshev polynomials of the first kind are jacobi polynomials with a = -1/2, b = -1/2. This is starting to remind me of the beta function.... With some slight tweaks, you can derive the other Jacobi polynomials (It's a little complicated to fit all the math into a youtube comment), but it works out that Legendre polynomials (orthogonal with no weight, so a = 0, b = 0) when applied to cosines are orthogonal realive to a weight of sin(x)^1. The pattern continues for the rest of the Jacobi polynomials. Chebyshev polynomials of the second kind (jacobi polynomial for a=1/2, b=1/2) are orthogonal to (1-x^2)^(!/2), but applied to cosines, they are orthogonal relative to sin(x)^2. So if a jacobi polynomial is orthogonal relative to a weight of (1-x^2)^a (simplifying to set a=b), when applied to cosines, it's orthogonal relative to a weight of sin(x)^(1+2*a), which is directly reflected in the relationship that polynomials have in the beta function. It ends up popping up in Bernstein polynomials too, which is obvious considering they're basically the type of function the beta function integrates, stuff in the form of (1-x)^n*x^(v-n). The beta function for polynomials in that form just ends up being binomial coefficients (n; v), (the actual coefficients are different, but it's equivalent due to symmetry). Each polynomial is normalized so that each member of a set of polynomials contained in v to all integrate to the same value. The sum of this same set of polynomials is just 1, and these properties makes them a "partition of unity"

[@subhrajitroy1477]

hey papa, i broke my neck trying to read the meme :'(

[@subhrajitroy1477]

JUST GOT ADDICTED TO YOUR VIDEOS....TODAY I SPENT 4 STRAIGHT HRS ON KZbin, WATCHING YOU

[@spacejunk2186]

>laughs in australia

[@subhrajitroy1477]

@@spacejunk2186 :)

[@tretolien1195]

Ohhh Flammy diden't you know you could just have used the advanced level 'Horseshoe mathematics' method and have gotten 100% in less than a minute

[@owenl3929]

HERE is what I was looking for Papa Flammy

[@emperorpingusmathchannel5365]

Damn that meme in the beginning. I don't speak Australian!

[@srinivasadireddi]

at 12:41, why do we need to multiply it with the det of jacobian matrix?

[@miro.s]

Awesome! I've got relaxed during your video.

[@jarogniewborkowski5284]

Did You already made video about Jacobians and how it can be used in integrals like in this movie You have used?

[@antronixful]

thank you papa... my girlfriend asked me for help with a programming task, which consisted in finding the zeros of a function of her choice, using numerical analysis ... you had to use 6 different methods, but the function was the important thing (hehehe), so I came immediately to the papa's channel looking for the sickest function... also put some mini-game in the in the script for the waiting time (i.e. for the memes) xd

[@jacoboribilik3253]

Nice definition of the Beta function. It is used in Bayesian statistics very often too.

[@frozenmoon998]

There is a movie called Close Encounters of the Third Kind, but this is better! It is called Eulerian Integral of the First Kind :)

[@wiloux]

that 3:13 integareale pronunciation is gold

[@MathematicsOptimization]

omg pops now that u mentioned it pls do some multivitamin calc like div, grad, curl, jacobi, stokes theorem and shit!!!

[@MrCigarro50]

Great video. Thank you.

[@PapaFlammy69]

glad you liked it! :)

[@leafbaguette]

I know you math bois don't like this but the physics notation of putting the d(dummy variable) right next to the integral sign makes it easier to not lose track of bounds

[@metuphys5611]

it looks better that way too

[@arbitrarilyclose]

Papa seems so happy at the start of the videos :)

[@fanyfan7466]

Finally! I love the Beta function

[@thecustomer2804]

*Papa Flammy makes new video* Me: oooooo *clicks*

[@fym4x7]

Me: I'm gonna watch a *B*orn KZbin recommendation: Me: fuck yeah

[@mathhack8647]

Exellent. Now I understand Beta Function. and I can solve. integral. of ln(cos(x)). in another way . Thanks

[@svenweiland3322]

Oh no KZbin recommended fresh toad walker's new video with "Zuschauer von Flammable Maths schauen sich diesen Kanal an"

[@svenweiland3322]

@@PapaFlammy69 hopefully just to troll him or at least get mad at him internally.

[@silentinferno2382]

Trig version? Coming soon? I haven't forgotten about the shirt.

[@athul_c1375]

papa can you name the book I can study these special functions

[@uva1312]

Papa, I think it would be really cool if you put out a video on the incomplete definition of the gamma function. No pressure haha, just a suggestion for the future. Keep up the amazing videos.

[@subhrajitroy1477]

Today PAPA became proud by receiving fan mail...for the kids who didn't know. BTW 9TH COMMENT PAPA!!!

[@mahmoudkhamis409]

4:26 yes that's right but in Quantum mechanics I don't :)

[@hassan010012]

Awesome!

[@insert_a_good_name_here4585]

Have you read 'inside interesting integrals' by Paul Nahin? Just curious :)

[@nanigopalsaha2408]

I have read it. Absolutely fantastic!

[@surferriness]

Halfway through the vid Didnt correct the t to Tau TRIGGERED

[@biswadeepchatterjee6074]

Papa flammy can u make a video on Jacobian determinants nd matrices plzzzz for ur fellow mathematicians

[@spacejunk2186]

Lol this boi thinks the Euler integral actually exists.

[@hacker2ish]

Pls explain why gamma is continuous

[@paulbucher4655]

Papa❤

[@michelkhoury1470]

Cheers papa Laplace :p

[@huhulili9021]

Mr Daddy, is it alright if I ask a question, what's ur view in learning applied math for comp science? I'm at a cross road on whether to do a degree in CS or in applied math

[@pacman7328]

If there's a beta function then where is the Alpha function?

[@vaualbus]

The finalproblem is does it exixt the fourier transform of the gamma function? And if so how to calculate it :)

[@karolakkolo123]

To my naked eye, the answer is no. Even if there is a fourier transform of the gamma function, it is most likely not a function itself, but a distribution. And that distribution doesn't look like it would be nice to handle. Just take a look at the positive reals with Im(z)=0, and the fact that it is non-periodic-like and monotonically increasing starting with the minimum between 1 and 2

[@s1ng23m4n]

excuse me, but I still do not understand, is the beta function purely derived from the gamma function?

[@ChanawerebiChanawerebi]

the beginning of this video..... Misophonia........... 😭😭😭😭😭

[@unknownknown347]

Papa you inspire me!!!

[@garogarabed6196]

11:20 the same Spiel lolll

[@yotty97]

Can't you give an intuition for the beta function? Like the gamma function is the continuous analog of the factorial function....so what is the intuition behind the beta function? EDIT: i just found out it's related to (and in a sense derived from) the binomial function - you should really have made a mention of this. Just like defining the gamma function in terms of a continuous factorial, it's extremely useful to be given a motivation for the beta function too

[@nicholasquiroga2861]

epic, thank you

[@PapaFlammy69]

:)

[@matheus_rml]

hey papa, I challenge you to solve the sum from 0 to infinity of 1/((4n+1)^2), this is actually a challenge that I recieved from my friends and I couldn’t solve

[@matheus_rml]

Flammable Maths holy shit that was fast

[@angelmendez-rivera351]

Matheus Ramalho Once you see how to solve it, you'll be mindblown. It's not very difficult, but you do need to be quite clever.

[@angelmendez-rivera351]

Flammable Maths The question is, can you solve the alternating version of that series? Namely, 1/1^2 - 1/5^2 + 1/9^2 - 1/13^12 + •••. It's much more challenging :)

[@etasyr]

One of the definitions of Catalan's Constant (which he already made a video on) is G = -⅛𝛑² + 2 * ∑ 0 to ∞ of 1/(4n+1)² Rearrange and ∑ 0 to ∞ of 1/(4n+1)² = ½G + ¹⁄₁₆𝛑² It would be nice were he to show how to get to that definition ^_^

[@matheus_rml]

Angel Mendez-Rivera of course I can’t hahahaha

[@bon12121]

NEW SUBSCRIBER!

[@PapaFlammy69]

Hi! :3

[@Mystery_Biscuits]

Video is 17:29 on the thumbnail, woo!!!

[@thesattary]

why you are so good?

[@matron9936]

Please solve Integral(ln(x)sec(x)dx)

[@angelmendez-rivera351]

Matron I think this is a non-elementary integral. This is because if you use integration by parts, you eventually integrate the antiderivative of secant, and I am fairly certain that one is well-known to be non-elementary.

[@thephysicistcuber175]

Trivial with horseshoe integration bruh

[@peterdriscoll4070]

Nice!

[@peterdriscoll4070]

@@PapaFlammy69 Is beta related to the reciprocal of x+y comb x = (x+y)!/x! y!

[@novanecros9145]

Papa I have failed you. i factorial sent me to the reflection formula which then sent me here and then I realized I haven't studied multivariable calc so I'm not ready for the awesomeness. I'll be back in two weeks, promise. :(

[@adithyar4282]

everyone will have one apple that is Adam apple

[@obaidurrehman2464]

Saying " cool ,cool " again and again 🤣🤣🤣🤣

[@JamalAhmadMalik]

Papa or papá?

[@dannygjk]

papa=adad ;) Kelly's Reflection formula.

[@ducksfan1018]

Yeah sex is cool and all but have you seen Papa Flammy destroy inteGERALS by the thousands

[@gdsfish3214]

When is the redpill alpha integral coming libtard?

[@holyshit922]

What about derivatives of Beta function fe Int((ln(cos(x)))^n,x=0..pi/2) t = -ln(cos(x)) -t = ln(cos(x)) exp(-t) = cos(x) -exp(-t)dt = -sin(x)dx exp(-t)dt = sin(x)dx exp(-t)dt = sqrt(1 - exp(-2t))dx dx = exp(-t)/sqrt(1 - exp(-2t))dt Int((-t)^nexp(-t)/sqrt(1 - exp(-2t)),t=0..infinity) Int((-1)^nt^nexp(-t)/sqrt(1 - exp(-2t)),t=0..infinity) Let f(t) = 1/sqrt(1 - exp(-2t)) and L(f(t)) = F(s) Our integral equals d^n/ds^n F(s) at s = 1 Lets calculate L(1/sqrt(1 - exp(-2t))) Int(exp(-st)/sqrt(1-exp(-2t)),t=0..infinity) u = exp(-2t) du = -2exp(-2t)dt du = -2udt dt = -1/(2u)du -1/2Int(u^{s/2}/(u sqrt(1-u)),u=1..0) 1/2Int(u^{s/2}/(u sqrt(1-u)),u=0..1) 1/2Int(u^{s/2-1}/(sqrt(1-u)),u=0..1) 1/2Int(u^{s/2-1}(1-u)^{1/2-1},u=0..1) L(1/sqrt(1 - exp(-2t))) = 1/2B(1/2,s/2) Int((ln(cos(x)))^n,x=0..pi/2) = d^n/ds^n (1/2B(1/2,s/2)) at s = 1 But how can I calculate derivative of Beta function

[@CDChester]

THAT GULP DOOOOE #ASMR

[@xdtidebringer5583]

Nice

[@razmakbazai3556]

wow

[@mrinalchoudhury2725]

Ons wakhan

[@kevind.shabahang]

cool :)

[@benjaminarias5193]

Gucci af you boi

[@sofianeafra6161]

Hey yen say 555 in Germany 😂😂

[@sofianeafra6161]

@@PapaFlammy69 oh shit ! Studying maths for pH.D is easier than pronouncing this word 😂😂

[@garykang3712]

Please don’t make a confusion with t and tau

[@michelkhoury1470]

Ummm I think I did it with the same way

[@kwirny]

Ok Papa,now i have enough from the gamma stuff :().