ONE NEAT PROOF! Deriving the EULER DEFINITION of the Gamma Function!
HTML-код
- Опубликовано: 1 май 2019
- Help me create more free content! =)
/ mathable
Merch :v - teespring.com/de/stores/papaf...
shop.spreadshirt.de/papaflammy
2nd Channel: / @rockhardwooddaddy
More Info: • Gamma Function Euler F...
Gamma Fct: • DERIVING THE GAMMA FUN...
Gamma Weier Form: • The Weierstrass Defini...
That one's going to be fun! Today we are going to prove, that we can express the Gamma Function in integral form as an INFINITE PRODUCT! At first we manipulate a simple factorial into a limit, and then prove, that both expressions are indeed equal! Enjoy! =)
My Website: www.papaflammy.engineer/
Flammy's subreddit: / flammybois
Twitter: / flammablemaths
Facebook: / flammablemaths
Got some time to spare? Make sure to add captions to my videos! =) ruclips.net/user/timedtext_cs_p...
That moment when Papa Flammy defines n as being strictly less than z and then takes the limit as n approaches infinity while keeping z finite... (I assume that there's a second case to the derivation in which n is greater than z that ends up with the same result at some point down the track, but still a bit slack :P )
but generally z is a complex number,so z>n is meaningless,untill we are speaking of the modulus
This reminded me a class in Materials Science when the teacher approximated sin(theta) by theta for very small angles, and then proceeded to integrate between 0 and 2pi. :-)
@arabicboi Yes you‘re right but than it also wouldn’t make sense to define n < z. Please correct me if I‘m wrong but I think this proof is not valid.
@@epicmorphism2240 I don't think it's meant to be an entirely rigorous proof, just sort of showing the general ideas
Rish There is a difference between rigorous and wrong…
I HOPE YOU AGREE WITH ME
*vague germanic threat*
"If you have one apple then you still have one apple" -one random flammy boi
10:01
About that n = [ 2/1 • 3/2 • 4/3 •…• n/(n-1) • (n+1)/n], shouldn't it be left with n + 1 after those cancellation?
Yes I Had the same question papa
That needs to be more clear. Just saying that it does not matter in limit would not be enough.
Well spotted @Silver-G
it should be n/n-1 in the end
It's a very good question. But see in the video by min 10:24. He changed (n+1)/n into 1+(1/n). In this version let n run to infinity. So you have n=n.
The idea of reetanshu is also remarkable. But you have to consider that this is the penultimane term in the equation.
When he started to taking the limit of n, you can asume z fixed (but z except -infinity or infinity) . For zn is a little bit more complicated, but I think it works too using horizontal truncation of lebesgue measure.
If z=n, I don't know.
He's fine :)
Besides the random groaning at 00:03 I have something to tell you and I hope you see this.
I am an engineering student whose relation with mathematics kind of became sour after entering university ( there are several reasons but I don't want to get into them). Although, I have no idea about how I discovered your videos, I'm very glad that I did. There are something in your videos that made me realise how interesting and enjoying math really is and I should not give up so easily due to my failures or succeses of others. I would like to write more but i don't want it to get more boring. I just want you to know that I owe you a lot. Thank you so much and please keep up with your videos :)
Where is the uncut version on your second channel my twink boi???🍄
When deriving the partial integral of the { t^(z-1)(1-t)^n }, it is much simpler to do it in case of n=3 as an example.
Then the resulting form clearly hints us about the form in case of n.
For the purpose of explaining , I think this way is better. Don't you think?
that was a seriously good video!thanks for the outstanding great work as always
1:55 You said n is strictly less than z. If you let n go to infinity, does that mean that this only works for z also going to infinity?
9:56 the way you explained it I got the idea that the final product in the brackets would leave us with just the "n+1" term, without it being divided by anything. Every previous term's numerator and following term's denominator will continue to cancel out, including denominator of (n+1)/n term. Leaving us with just "n+1". Someone pls prove me otherwise cause I can't properly continue watching the video without resolving this misunderstanding.
Lmao, I'm crossing with the same question. I've started studying the gamma function and saw a similar process but the product is set equal to (n+1) not n. I've tried to make sense out of it, but since I couldn't, I stuck with the other method.
8:24 How about the Ganna function?
I once read this on a book called "Funzioni Speciali" From L. Gatteschi. Never found that book again.
Lmao literally the funniest math tutorials! Bravo! Laughed out loud at 7:26 "we are just going to take the limit of this crap"
Actually the integral t^(z-1)(1-t)^n is a beta function with parameters z and n+1. beta function can be rewritten in terms of gamma functions as a fraction gamma(z)gamma(n+1)/gamma(z+n+1). I think this relation was used on the channel, but I don't remember in what video.
7:30 another important theorem to mention is that you have finitely many 1 limits, otherwise you're doing an advanced e.
Papa’s videos are the best way to start the long ass day at work!
And the Gamma function as derived by DADDY EULER? Even better!
simply awesome definition for the Gamma function!
I have a doubt, when you take the limit as n goes to infinit, shouldn’t you just take it on the last term? Because Its like a progression. Sorry for my bad english.
Was that a 3Blue1Brown-roid?
Ganz prima Video!! Danke!!
9:57 cancelling everything out, you end up with n+1 ...? What did I miss?
You're right.
Just for fun can you do a video on the integral of sec(x)tan^2(x). It is beautiful because you have to evaluate sec^3(x) which involves coming back to the original integral. Or you could make it a bit harder by doing sqrt(x^2+1).
but in the initial series n
Where’s the link for the integration by parts uncut version?
That yoke at beginning is very funny😂.
I suggest you a question please answer
gamma (n + 1/2) as a product what is equal to? how to write with the product symbol
8:50 "I wanna play more with this junk over here"
I bet you do, huh?
Shouldn't the product go from k=1 to n-1 (not n)?
The integral at 17:12 is oddly similar to beta function...I think you could have used it's relationship with gamma function to express the integral in terms of factorials...
Priceless :) The various definitions of the gamma function just look too badass not to be derived
Why is the Gamma function used for the ao term of the bessel function of the first kind?
The Gamma function is cool and everything, but are you ever going to do a video on the Borwein integral? :)
For π expansion from '1' to 'n' while expressing 'n' as a finite product you can only pull this upto n-1 as the upper limit then why and how did you go for n as the upper limit. Plz explain.
@@PapaFlammy69 but still, there limit is applied on that basis it's fine to comprehend but here it's just doesn't go through. Help me with this if you can.
How can you write a complex number factorial as a real number factorial times's (n+1) .... z ??????
here n
what did you say from 12:27 to 12:29?
im sorry but, why you can cancel out the z/n as n goes to infinity?, i mean, if n goes to infinity, doesnt that mean that z also goes to infinity?
When you go from (n/(n-1)) to (n+1)/n is where I stopped the video. There should be more background as to how you can justify using the fact that n=n can be turned into n=n+1 which of course is not an equality.
Saw one of your snack videos and love all of your content. Was curious how someone else did the i! I remembered that you did a video on it because the video I saw from someone else resorted to approximation rather than a closed form and came here by your link on that video.
I made a complementary video on that actually!
Find out the numerical value of Γ(1/3), Γ(1/5) ? Or Γ(1/3) =?, Γ(1/5) =?
imo the nth integration by parts was more interesting than the Euler definition of the gamma function lol
But shouldn't z also approach infinity, as n approaches infty?
Yeah, I have the same question
That's why í disliked this video
@@Dionisi0 low brain
He said n < z for the sake of simplicity, since it would make breaking the product the way he did easier, but actually, there is no need for n to be strictly smaller.
Please create a video where u prove gauss's multiplication theorem pleaaaaas
Thank you folk your video was very usefull, I hope people agree with me
I agree
Just a comment to increase papa's popularity
Actually it will be wrong to demand that n will be strictly less then z, because you want to take a limit of n to infinity and don't change z.
the limit of ab is not necessarily equal to the limit of a times the limit of b?
also 1/2 * 2/3 * ... * (n+1)/n = (n + 1)/1 = n + 1 which isnt equal to n?
EDIT: think it was supposed to be (n+1)^z on the LHS,
EDIT 2: gotta finish the vid another time, gotta go to bed
Will you proof "classical" form of gamma function i.e it's integral representation
@@PapaFlammy69 thanks :D
You need some Hagoromo chalk
I have a doubt at 9:47 n=(2/1)*(3/2)*(4/3)*..........*(n/n-1)
nononono 10:05 idk if you realize but you just did n=n+1, which is technically fine for infinitives but is very dubious
Ryan Roberson How is it dubious? You literally just said it's fine for infinities. Pick one.
我看看这个视频到底能不能让我彻底攻克gamma function
how the fuck can you take \(n
ightarrow\infty\) if n in an integer less than z??
"In the real numbers, shit is Abelian." - Flammable Math 2019
Bro how e to the power -x is 1-x/n to the power n
Thats basically the definition of e
do the limit of (1+1/n^)^n n--> infinity and youd get e
it is very famous and you could see the proof online
SHOT CLOCK CHEESE!!!
Shot clock cheese
You’re building up to something, I know it.
OOF for the first time I gotta get a pen and paper and try it myself to be convinced it really works :(
Класс. Спасибо
Some geometric insight please. Please.
Euler Smells like a GOD🗿🗿🗿🖤🖤🖤
La matemática es independiente del idioma.
SHOT CLOCK CHEESE
Spicy
老子还是没懂
For me something is wrong in factorial proof! Sorry!
Te amuuuuuuuuuuuuu
too bad
On a related note: If you define,
P(x) = integral Γ(x)dx
then integrate P(x+1) by parts, you get
integral P(x) dx = xP (x) − P (x + 1)
All succeeding integrals of P(x) are also algebraic, as are polynomial forms, for example the all integrals of the polynomial
sum(m,n)[x^m P^n(x)]
can be expressed as a polynomial of a similar form.
The same is true if you define
Q(x) = integral (1/Γ(x))dx. As you know, 1/Γ(x) is well-behaved.
The only thing is, that I don't know what use these functions have.
Last part i.e when you just changed the form of n^z and prdouct of (z+1)(z+2)...(z+n) is not so clear.
Also, you must try to be somewhat official....(use good language).
You talk german? I mean you kind of sound like one.
@@PapaFlammy69 Ich wusste es, auf welcher Uni bist du?
@@PapaFlammy69 Ich kann mich nicht entscheiden, Bonn oder Aachen. Will im Wintersemester anfangen.
@@PapaFlammy69 SUNY Potsdam? :-P
Ist Bonn geworden :), viel zu tun aufjedenfall.
SHOT CLOCK CHEESE!!!
This is a Monky
SHOT CLOCK CHEESE
SHOT CLOCK CHEESE!!!
If I had a nickel for every time I read this today, I would have 5 nickels, just enough to afford the life of people who write that shit
SHOT CLOCK CHEESE
SHOT CLOCK CHEESE
SHOT CLOCK CHEESE
SHOT CLOCK CHEESE!!!
Wtf does that mean
beta slay My left stroke just went viral
Marko Rezic Right stroke put lil' baby in a spiral