This very day a 4 year friend of mine and I had to seperate ways, he was having other people to write a note or something similiar to him as a memory. And since I'm a mathguy too, along with the note, I drew the gamma function and explained it. Seeing this video getting uploaded the same day is beyond amazing.
I've noticed that in univerisities, even the "good" universities, in math courses they never discuss "why" certain results are they way they are, they just teach and are concerned with the proofs and rigour, which is reasonable, but discussing the why actually can give you a big insight in the problem an its underlying aspects, as well as it is a good way of facing problems in general
@@marcushendriksen8415 of course. But that is very tautological. Properties or thoerems arguably don't "have purpose", but asking "why" in the end is a way of developing intuition. And that is very important, in my opinion
Bro, that was just brilliant. I've never seen something like this before. I saw the proof of the fact, that Gamma is an extension to the factorial immediately, just integrating by parts in my mind. But I've never seen how to DERIVE this representation. Thank you, sir. This is gorgeous.
Awesome! Learned it at an even deeper level! At 58, I still marvel at the wonder of math. Can you do one on a function that provablely can NOT be generalized next?
Watching this the day before defending my bachelor's thesis about the Gamma function, Bohr-Mollerup's theorem and Stirling's approximation formula! Getting me very excited to finally share my work at uni :)
Just to mention, the Gamma function was found by the brother (Harald) of Niels Bohr and Johannes Mollerup. The Gamma function is the only function for x>0 with ln ( f ) convex ( f is ln convex) , f(1)=1 and f(x+1) = x f(x).
Sorry but that is quite wrong. The gamma function was discovered by Daniel Bernoulli. Euler also independently found it at the same time as well but Daniel actually came up with a solution first.
Fantastic proof. I have a question: why not simply define Γ with t^x inside the improper integral, instead of t^(x-1)? In that way, we would have Γ(x) = x Γ(x-1) and then we could define: n! := Γ(n) = n Γ(n-1) = n (n-1)! This would remove that stupid -1 inside the exponent of t in the definition of Γ, and the definition n!:= Γ(n) seems much more natural than n! := Γ(n+1).
That is the pi function. Capital pi. I agree that is the more natural function to take, which is why I use the pi function instead of the gamma function.
Multiply top and bottom by (-1)^{n+1}. On the top you get (-1)^{n-1} * (-1)^{n-1} = (-1*-1)^{n+1} = 1 and on the bottom you get (-1)^{n+1} * (x)^{n+1} = (-x)^{n+1}
8:54 I'm sorry am I just dumb or am I missing something 😭 the limit is as r goes to infinity, how would the exponential disappear if it's e^rx as r -> ∞?
This is so much good and great video to learn how does this gamma functions come. Today I understood the gamma functions. I love your videos. keep making such mystery behind the math. Thanks.
With the ending, 10:42, where you say we've only proved this for natural numbers, isn't the Gamma function what we use to define non-integer factorials in the first place? So there's no need to prove this for other values - we wanted a way to generalise the factorial function outside of the natural numbers and the Gamma function is the way to do that since plugging in n values that aren't positive integers makes sense in the context of the integral, while it doesn't really make sense in the context of our current notions of the factorial function at this point. Great derivation by the way, awesome to see this built from the ground up though it did get a bit confusing - perhaps you could have made it a little clearer about why we introduced the new variable t, but I managed to understand a majority of it on the first watch so I think you explained it very well :)
wow, amazing. Best explanation i've seen so far as somebody who never really understood where it came from. The only thing I'm kinda uncomfortable with is the cheeky integral slip in the final couple of steps. Could anyone explain to me how that works?
@@beanzthumbz we have no definition for factorials for real numbers without the gamma function n!=Γ(n+1) is a theorem for natural numbers x!=Γ(x+1) is a definition for real numbers. So there is no way to prove this for real numbers as it is a definition
@@Your_choise makes sense. But how would you prove that the gamma function is a good fit for extending the definition to the reals? As in prove that the graph is smooth and well behaved between the naturals, not some wiggly mess
I've been interested for years in efficient ways of calculating the gamma function of non trivial values to arbitrary precision. It looks like extending the Stirling formula to include as many terms as required for a desired accuracy is still the accepted way of doing this but recently I've been experimenting with continued fraction representations.
I am sorry to ask but at 10:24 you said that derivate of e^xt n time according t is t^n .e^xt but let's take an example : (e^tx)' = t.e^tx (e^tx)'' = (t.e^tx)' = (t^2 +1).e^tx not = t^2.e^tx Did I make a mistake because I don't understand
He is taking the derivative with respect to t, but the first derivative you took was with respect to x. d/dt(e^(xt))=xe^(xt). Taking the second derivative with respect to t gives x^2e^(xt), and in general the nth derivative with respect to t would be x^ne^(xt)
mmm my ɛ cents here: since x0) and since e^{xt} is "well behaved" on the real axis, it won't show any infnite behavior anywhere, so we can (¿¿¿???) rest asured any derivative of the function will also be bounded and well behaved and then the integral can converge uniformly to a function of x. But I can be (and I bet I m) totally wrong with this intuitive reasoning :)
I never learned the power rule, so I probably didn't do any of the other things that I would have done in that lesson. I'm from the UK and I stopped maths at school at GCSE. I also don't know calculus.
Before you can ask if there exists any x in some S such that f(x) = y for some y in some T, you need to actually define what f is. Otherwise, the question is malformed, and thus, incoherent. The notation you are using for f in this case implies that f is a so-called "absolute value." An absolute value f is a function from an integral domain D to the real numbers R, such that: •for all x in D, f(x) >= 0 •for all x in D, f(x) = 0 if and only if x = 0 •for all x, y in D, f(x•y) = f(x)•f(y) •for all x, y in D, f(x + y) =< f(x) + f(y) What the definition implies is that, regardless of what D is, for all x, f(x) >= 0. You are asking to find some x in some D such that f(x) = -1, but this would then require -1 >= 0, which is impossible. Therefore, for all D, there is no x in D such that f(x) = -1.
hope to see mathematicians cover some probability things like nPr nCr nIIr not playing with some BEAUTIFUL numbers....easiest way and every different interpretation possible, not just simply n CHOOSES r.
Look into the pi function (capital pi). It is exactly that. I think the way the gamma function tends to pop up somehow makes it the “nicer” choice, despite this relation to factorials having the shift by 1.
10:42 "We proved it for natural numbers only, not all the numbers we were looking for...but...it does happen to work out" My boy, you blew it. No such thing as "it does happen to work out" regarding rigorous mathematical proofs. What a letdown! You blew it and I wasted almost 12 minutes on this video. I got duped! You could have said it in the beginning of the video, but you were a sneaky little bastard and said so only in the last few minutes of the video.
What was the point of that though? You proved the gamma function only for positive integers? So youve extended the factorial to the positive integers. Congrats, you did a lot of work to make the factorial more complicated. I am being playfully sarcastic here, but I legit dont see a purpose... without proving an extension to the non-integers you dont really have a gamma function at all.
First of all: There is no good reason to have it defined as you expected, but the mathemations decided to start with having the value undefined for Gamma(0) and start with Gamma(1)=1, so that for all positive x>0, the Gammafunction delivers a positive value, i.e also for Gamma(1/2) etc. With that you get the functional equation as Gamma(x+1)=x Gamma(x) and you have Gamma(x)>0 for all x>0 Notice: on the negative side the function flips from negative to positive and vice versa between negative integers. With your definition you would shift this behaviour and the first flip would be between 0 and 1. It's somehow nicer to have it defined like the mathemations did, but they could also have used your definition, but then everything shifts by 1. Look at the Graph of n! and think over how to complete the gaps in between.
That is the pi function you are asking about (capital pi). I personally prefer to use that as it seems to be the more natural function in my opinion, but the gamma function shows up everywhere as it is, so I guess it’s somehow the “nicer” function given the contexts it shows up in
10:12 he doesn’t say it because it’s besides the purpose of this video, but we in fact just found the Laplace Transform of tⁿ for n ∈ ℕ, so generalising it to n ∈ ℝ, we just found the Laplace Transform of tⁿ for n ∈ ℝ.
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This very day a 4 year friend of mine and I had to seperate ways, he was having other people to write a note or something similiar to him as a memory. And since I'm a mathguy too, along with the note, I drew the gamma function and explained it. Seeing this video getting uploaded the same day is beyond amazing.
I've noticed that in univerisities, even the "good" universities, in math courses they never discuss "why" certain results are they way they are, they just teach and are concerned with the proofs and rigour, which is reasonable, but discussing the why actually can give you a big insight in the problem an its underlying aspects, as well as it is a good way of facing problems in general
i prefer to discuss the why first find something that works, and concern with the rigour later
Proofs tell you the 'why' though?
@@HassHansson you gotta know why before you do it. otherwise your just infinite monkey theoreming over stuff, and thats not productive
The "why" is "because they work."
@@marcushendriksen8415 of course. But that is very tautological. Properties or thoerems arguably don't "have purpose", but asking "why" in the end is a way of developing intuition. And that is very important, in my opinion
Bro, that was just brilliant. I've never seen something like this before. I saw the proof of the fact, that Gamma is an extension to the factorial immediately, just integrating by parts in my mind. But I've never seen how to DERIVE this representation. Thank you, sir. This is gorgeous.
Awesome! Learned it at an even deeper level! At 58, I still marvel at the wonder of math.
Can you do one on a function that provablely can NOT be generalized next?
That was sick!
It just reminds me how creative these mathematicians are. This is math: being creative enough to come up with crazy new things
Watching this the day before defending my bachelor's thesis about the Gamma function, Bohr-Mollerup's theorem and Stirling's approximation formula! Getting me very excited to finally share my work at uni :)
Thanks
Thanks so much!
Just to mention, the Gamma function was found by the brother (Harald) of Niels Bohr and Johannes Mollerup. The Gamma function is the only function for x>0 with ln ( f ) convex ( f is ln convex) , f(1)=1 and f(x+1) = x f(x).
Sorry but that is quite wrong. The gamma function was discovered by Daniel Bernoulli. Euler also independently found it at the same time as well but Daniel actually came up with a solution first.
@@larzcaetano yes but they did not prove that the Gamma function is the only function that has the listed requirements.
@@JustNow42yes but your original comment wasn't about proof, but discovery...
Fantastic proof. I have a question: why not simply define Γ with t^x inside the improper integral, instead of t^(x-1)? In that way, we would have Γ(x) = x Γ(x-1) and then we could define:
n! := Γ(n) = n Γ(n-1) = n (n-1)!
This would remove that stupid -1 inside the exponent of t in the definition of Γ, and the definition n!:= Γ(n) seems much more natural than n! := Γ(n+1).
That is the pi function. Capital pi. I agree that is the more natural function to take, which is why I use the pi function instead of the gamma function.
this was honestly a mind blowing explaination, amazing job
That's actually a very great video 🤙🏻
I'd like to see more of the same topic
Nice work 👏🏻
it was very good induction to derive Gamma function!
many thanks
At 6:06 how did you use properties of exponents to get to the bottom line from the line above?
Multiply top and bottom by (-1)^{n+1}. On the top you get (-1)^{n-1} * (-1)^{n-1} = (-1*-1)^{n+1} = 1 and on the bottom you get (-1)^{n+1} * (x)^{n+1} = (-x)^{n+1}
My favorite video of yours!
8:54 I'm sorry am I just dumb or am I missing something 😭 the limit is as r goes to infinity, how would the exponential disappear if it's e^rx as r -> ∞?
If x -∞ and e^(rx)->0
@@andrewkarsten5268 yup that's the detail my mind was missing. Thank you for pointing that out to me again
@@mae_lia I was confused on this too. Thank you for clarifying.
4:00 been watched his old videos I thought he was going to say "let's use the gamma function" 💀 😂
I absolutely loved it!! So simple...so utterly simple....
Integration by parts is also intuitive, every time you integrate get sequence n(n-1)(n-2)(n-3)...
This is so much good and great video to learn how does this gamma functions come. Today I understood the gamma functions. I love your videos. keep making such mystery behind the math. Thanks.
With the ending, 10:42, where you say we've only proved this for natural numbers, isn't the Gamma function what we use to define non-integer factorials in the first place? So there's no need to prove this for other values - we wanted a way to generalise the factorial function outside of the natural numbers and the Gamma function is the way to do that since plugging in n values that aren't positive integers makes sense in the context of the integral, while it doesn't really make sense in the context of our current notions of the factorial function at this point.
Great derivation by the way, awesome to see this built from the ground up though it did get a bit confusing - perhaps you could have made it a little clearer about why we introduced the new variable t, but I managed to understand a majority of it on the first watch so I think you explained it very well :)
wow, amazing. Best explanation i've seen so far as somebody who never really understood where it came from. The only thing I'm kinda uncomfortable with is the cheeky integral slip in the final couple of steps. Could anyone explain to me how that works?
Also, how do you prove this for all real numbers? Since you stated at the end it only works for naturals :(
It is an special case of the Leibniz integral rule! en.wikipedia.org/wiki/Leibniz_integral_rule
@@sandromauriciopeirano9811 nice one, thank you. Any idea about how to generalise it to the reals as well?
@@beanzthumbz we have no definition for factorials for real numbers without the gamma function
n!=Γ(n+1) is a theorem for natural numbers
x!=Γ(x+1) is a definition for real numbers.
So there is no way to prove this for real numbers as it is a definition
@@Your_choise makes sense. But how would you prove that the gamma function is a good fit for extending the definition to the reals? As in prove that the graph is smooth and well behaved between the naturals, not some wiggly mess
I've been interested for years in efficient ways of calculating the gamma function of non trivial values to arbitrary precision. It looks like extending the Stirling formula to include as many terms as required for a desired accuracy is still the accepted way of doing this but recently I've been experimenting with continued fraction representations.
Best video about gamma function
Good to see you back with more calculus content :)
This is a really good follow up to a similar video by "line that connects". Thank you.
I am confused. How did you get from 5:32 from x's to n's. I am so lost.
_Always left me wanting more!_ Me too!
Ah yes 🔥😁
finally a satisfying explaination
I am sorry to ask but at 10:24 you said that derivate of e^xt n time according t is t^n .e^xt but let's take an example : (e^tx)' = t.e^tx
(e^tx)'' = (t.e^tx)' = (t^2 +1).e^tx not = t^2.e^tx
Did I make a mistake because I don't understand
I am sorry i understood i was a fool
He is taking the derivative with respect to t, but the first derivative you took was with respect to x. d/dt(e^(xt))=xe^(xt). Taking the second derivative with respect to t gives x^2e^(xt), and in general the nth derivative with respect to t would be x^ne^(xt)
But what is the benefit of generalizing factorial? What is the interpretation of negative and fraction factorials?
Yes!!! You finally returned to calculus!
Great video had never seen this insight.
5:13 - Bird
Gamma function is basically an analytical continuation of the factorial function. So thats is why it generates factorials.
9:47 I pray for uniform convergence
mmm my ɛ cents here: since x0) and since e^{xt} is "well behaved" on the real axis, it won't show any infnite behavior anywhere, so we can (¿¿¿???) rest asured any derivative of the function will also be bounded and well behaved and then the integral can converge uniformly to a function of x.
But I can be (and I bet I m) totally wrong with this intuitive reasoning :)
On that last point, does that technically make the gamma function an analytic continuation of the factorial function?
I never learned the power rule, so I probably didn't do any of the other things that I would have done in that lesson. I'm from the UK and I stopped maths at school at GCSE. I also don't know calculus.
Great video bro, thanks!💪💪💪
Dang, I’m gonna have to rewatch this video quite a bit before I’ll be able to get the hang of this gamma function thing down…
Is there any integral to interpolate superfactorials?
en.wikipedia.org/wiki/Barnes_G-function
Thank You
man you are aweasome ... can you tell me how do you make your videos
it is really good to understand this video
really really great!
Bravo!
The question is if we solve the functional equation f(x+1)=xf(x) do we get f = gamma
This equation has infinitely many solutions. You need additional conditions to get the Gamma function as the unique solution.
This proof just blew my mind!
Love your videos
This is Brilliant!
Very nice!
cool, ty )
If x^2 = -1 then x=√-1 or i
if |x|=-1 then what is x
Can you explain this
I couldn't find any video about this
Before you can ask if there exists any x in some S such that f(x) = y for some y in some T, you need to actually define what f is. Otherwise, the question is malformed, and thus, incoherent. The notation you are using for f in this case implies that f is a so-called "absolute value." An absolute value f is a function from an integral domain D to the real numbers R, such that:
•for all x in D, f(x) >= 0
•for all x in D, f(x) = 0 if and only if x = 0
•for all x, y in D, f(x•y) = f(x)•f(y)
•for all x, y in D, f(x + y) =< f(x) + f(y)
What the definition implies is that, regardless of what D is, for all x, f(x) >= 0. You are asking to find some x in some D such that f(x) = -1, but this would then require -1 >= 0, which is impossible. Therefore, for all D, there is no x in D such that f(x) = -1.
Great explanation.
What kind of bird did I hear in the background?
hope to see mathematicians cover some probability things like nPr nCr nIIr not playing with some BEAUTIFUL numbers....easiest way and every different interpretation possible, not just simply n CHOOSES r.
Incredible !
Why use gamma of n+1 = n! Why not gamma of n = n!?
Look into the pi function (capital pi). It is exactly that. I think the way the gamma function tends to pop up somehow makes it the “nicer” choice, despite this relation to factorials having the shift by 1.
very nice video man
Finally, someone had to this
1:42
Oh god that's annoying!
But using gamma function, I can't find factorial of 1/3
I’m just a 4th grader and it was ok but you need to learn lots of algebra and etc
This is beautiful
Define 0^0 please
1
Hi Bri!
Very nifty!
This is kinda nice
Or just use integration by parts + induction
“Intuitive”
X factorial
holy shit that was good
Bro your the guy on wizeprep
Hello, world!
Awesome :)
What are you saying 😩😩😩😩
use self.wait(1) and cut the manim animation a bit later than you do, I've seen a lot of incomplete animation at most of the end characters
The gamma factorial for most cases goes to either infinity or -infinity if you agree 👇🏻
Oi Vi que vc ia para B po e um abraço pra 🎉⅕
Dude, am a seventh grader
Well, why are you here then?
-1!=-1
Nah, you apply the factorial to the -1.
Like this: (-1)!
(i)!
10:42 "We proved it for natural numbers only, not all the numbers we were looking for...but...it does happen to work out"
My boy, you blew it. No such thing as "it does happen to work out" regarding rigorous mathematical proofs.
What a letdown!
You blew it and I wasted almost 12 minutes on this video.
I got duped!
You could have said it in the beginning of the video, but you were a sneaky little bastard and said so only in the last few minutes of the video.
Apply IBP and voilà, no need to whine
5:18 It should be -2.1x^-3
What was the point of that though? You proved the gamma function only for positive integers? So youve extended the factorial to the positive integers. Congrats, you did a lot of work to make the factorial more complicated. I am being playfully sarcastic here, but I legit dont see a purpose... without proving an extension to the non-integers you dont really have a gamma function at all.
but why is gamma defined as Γ(x+1)=x! and not Γ(x)=x! ?
First of all: There is no good reason to have it defined as you expected, but the mathemations decided to start with having the value undefined for Gamma(0) and start with Gamma(1)=1, so that for all positive x>0, the Gammafunction delivers a positive value, i.e also for Gamma(1/2) etc.
With that you get the functional equation as
Gamma(x+1)=x Gamma(x)
and you have
Gamma(x)>0 for all x>0
Notice: on the negative side the function flips from negative to positive and vice versa between negative integers. With your definition you would shift this behaviour and the first flip would be between 0 and 1. It's somehow nicer to have it defined like the mathemations did, but they could also have used your definition, but then everything shifts by 1.
Look at the Graph of n! and think over how to complete the gaps in between.
That is the pi function you are asking about (capital pi). I personally prefer to use that as it seems to be the more natural function in my opinion, but the gamma function shows up everywhere as it is, so I guess it’s somehow the “nicer” function given the contexts it shows up in
Г (x) = [(Пи ctg(Пи/2^x))^x]/2x
if, 0
Thanks for ending my 5 year journey on finding a derivation for the Gamma Function. 🫡
Amazing video!
10:12 he doesn’t say it because it’s besides the purpose of this video, but we in fact just found the Laplace Transform of tⁿ for n ∈ ℕ, so generalising it to n ∈ ℝ, we just found the Laplace Transform of tⁿ for n ∈ ℝ.
-1!=-1