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extending the factorial (the Gamma function & the Pi function)
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- เผยแพร่เมื่อ 14 ม.ค. 2018
- The usual definition for a factorial only works for positive whole numbers, but how can we take the factorial of any number? Here we will discuss the Pi Function which is defined in terms of an improper integral and it is also the cousin of the Gamma function. I also show the properties of the Pi function, which helps to extend the factorial. As a bonus, I also show why zero factorial is equal to 1.
Read more: en.wikipedia.o... ,
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Fermat, on proving his Last Theorem: 17:30
Hahaha nice one!
This is my favorite comment on youtube
Interesting proof ... as soon as you find it for yourself. (If a proof is never found, is it still interesting? If a tree falls in the forest when I'm not there, does it make an interesting sound? But interest is in the mind/eye/ear of the beholder, so is it in the ... .)
@@danielreed540 Very meta
@@danielreed540 r/iamverysmart
"-Hey mate can you tell me what's the factorial of 1?
-Yeah sure *pulls out integration formulas and l'Hopital's rule*"
the answer seems near to 2 that is near to 3 that is π
But it's near 2. Which mean it's e
You don't need the gamma function for that. You can figure that out just by using the two chosen properties listed in the beginning although this is only valid for whole numbers (which 0 thankfully is) which is why we needed the gamma function to extend the factorial (or rather some of its properties, which is actually what's happening here)
And after all that, so what if you need to apply by parts and L'Hopital? They're not _conceptually_ hard even though it might be a tenuous task to apply them. There's a big difference between something being logically hard and something being manually hard
the hospital
@@pbj4184 yo dude it's a joke
It is incredible what can be done with Euler's number. As time goes by you really begin to appreciate that number more than older societies appreciated pi
e^t never dies
87392739233935 637 is a lot
But he does phone home
Lol
Partial derivative kills e^t
ln(e^t) t=0 dead
Why is Gamma more popular and used than Pi? Pi seems more logic if you want a function that extends factorial.
The guy that invented the gamma function for some reason chose to make it off by one for some reason, and it became the most popular
Pi was probably done later
"That box means a lot to us" 😂 I died.
Bryce Lunceford hahaha. It does! The box is proofs just like +C is to integrals!
I didn't understand the import of the box?
@@trogdorbu same here but from what I understand it should be put when you end a proof of something 🤔
Black box good. Red box bad 😁.
@@protondecay4607 yes it is
Teacher: "Can you find a function so that f(1)=1 and..."
"a million brain cells pops up at once on your head"
That’s surprisingly only ~0.001% of your brain mass.
It seems to me that you don't need the gamma or Pi functions to show that 0! = 1. You just need the two definitions you gave.
1! = 1
n! = n * (n-1)!
Plug in 1 for n, you get
1! = 1 * (1-1)!
1 = 1 * 0!
thus *1 = 0!*
You have to be careful, though, because you can't just apply those formulas willy-nilly. Γ(1) = 0! = 1, but if you try to argue that since Γ(s+1)=sΓ(s), Γ(0) = 0 * 1 = 0, you run into trouble, because in actuality Γ(0) = ∞
Hi. You seemed to have made an error in your explanation:
Γ(s+1)=sΓ(s)
Set s=0
Γ(1)=0*Γ(0)
It is clear that one cannot decide the value here because division by 0 is undefined.
I think the point was calculating the integral rather than 0!.
Different points lay here.
@@Jodabomb24 doesn't gamma(s+1) = (s+1)•gamma(s)??
I am so happy you are doing this! Ive looked online for a reasonable way to understand the factorial function outside of just positive integers and have found nothing so far except this!
Yup but it doesn't really explain anything about why choose such a function? Where does it come from?
Patrick Apom You can prove that the Gamma-function is the only logarithmically-convex function interpolating the factorial.
Gamma function is one of the most popular functions, and is used extensively in evaluating various other integrals...
+Flewn: It doesn't really EXPLAIN much :q
+Arsh Khan: Neither does that.
Expressing those functions as integrals is putting the cart before the horse. A better way is to study the history of how the Gamma function has been discovered. Euler stated it for the first time with an infinite product, not an integral, which was much more close to the definition of factorial for natural numbers (which is also a product). Only then, when he tried to calculate its value for half-integers, starting from `(1/2)!`, he noticed that it results in Wallis's infinite product for `π/2`, and this gave him an idea that circles might be involved, so he switched to quadratures of the circle, trying to use trigonometric integrals to calculate the area of the circle. And this led him to the integral form used today. (Well, almost: it was closer to the Pi function mentioned in the video; the Gamma function with its "shift by 1" discrepancy is due to Legendre.)
Yes and no. People have been experimenting with x²e^(-x) etc before, so they might have noticed some stuff. Also there are other ways to express the Gamma-function. When we proved Bohr-Møllerop (the explicitness of the Gamma-funtion) in our last homework we also happened to show the Gaußian-Limit form of the Gamma-function (which could be the original Gamma-function maybe). There is also the Weierstraß one which uses Euler-Maccharoni-constant. You have to consider that it took decades for some of the greatest mathematicians out there to find all these crazy identities.
Around 12:00 you're discussing using L'H n times to kill the term but the whole point of this exercise is to create a function when n isn't an integer. If n isn't an integer in this step, you can't apply L'H n times to get a factorial like you say. Really you're getting n*(n-1)*... until the t term moves to the denominator, then you get a constant divided by infinity which does have the limit 0. Minor technical point. I love your love for math please never stop!
pure mathematics is the most beautiful subject according to me
I agree brother next to physics, for me its the most thrilled sub ever
@@eboian_x6522 This exact subject is being taught to us in physics and I’m only a 2nd year student. It’s not as ”pure” as one might think.
We are not playing hangman
Or we are?
**Vsauce music starts playing**
Euler again of course :)
Usually these kind of function are "deducted" by reasoning about "what you want" (as RedPen stated clearly in the video) and "which function is more suitable to fulfill the requirements".
Usually e^x comes up everywhere because of his extraordinary properties.
Well done RedPen!
Dude, you're such a good teacher! I never fully got why this worked until now.
If you want more, here is the wikipedia page. Wise words
of a wise man.
@@kanishk9490in wise pen.
This is fantastic. I am so glad I found this channel. Kelsey's videos and Mathologer are terrific, but the best way to explain math is to walk through it step by step on the board. I've looked at the gamma and pi functions on Wikipedia and the bit with x and t in the integrals had me stymied, but here at the end of your video when I looked back at the first integration I had none of my earlier confusion-- the x's role was immediately obvious, and I never even thought about it during the entire video. Looking at a page full of calculations it takes a lot of work to decode the operations and relationships. But watching it unfold in front of you is a cakewalk. LOL It's the next best thing to homework.
skoockum Who is Kelsey?
@Deepto Chatterjee: Former host of PBS Infinite Series.
PBS Infinite Series
You'll see this recommendation all over the comments on this channel, but 3blue1brown is another terrific math channel which uses clever and well-executed visuals to bring complicated concepts within range of your intuition
intresting fun fact:- th-cam.com/video/YIs3th01NV0/w-d-xo.html
You are the best math channel on TH-cam. 3blue1brown is great and all but you get much more into the nitty gritty. Thanks man.
You are one of the great youtubers.
And very good maths teacher I like.
Thank you!
I never laughed so hard while watching a mathematics video on TH-cam - 16:06. Thank you so much for the video man, I've been trying to understand the Gamma function for so long and your video explained it flawlessly.
Finally! But how dd they come up with the Pi and Gamma functions?
Hmm, you may have to ask Euler or Gauss for that.
I guess they saw how we can use IBP on those integrals and resulted some kind of factorial properties... I am not entirely sure tho...
Here is a nice summary of the actual history:
www.maa.org/sites/default/files/pdf/editorial/euler/How%20Euler%20Did%20It%2047%20Gamma%20function.pdf
As is often the case, the historical context and development is not very useful to the modern learner. People of different eras have different perspectives.
Somtimes the historical context is really important to understand how things came up and avoid some circular reasoning.
I'm thinking particularly of the log and exponential function, which each have many different definitions.
Experimentation, wealthy people had a lot of time on their hands back then.
Think about how many times people have integrated x^2 e^x for calculus exams or x^3 e^(-x), its maybe not so hard to imagine people trying to generalize it and find properties.
parsenver [Wealthy] people don't experiment, they [don't] need to ¡!
This was juat the explanation for
both these functions ive been looking for, thank you.
原來pi function 和 gamma function 這麼相近! very clear explanation!
Great!
How Gauss did come up with this anyway? And why is the gamma using x-1? Why not using the PI function directly?
Patrick Apom. I was wondering why Gamma is most famous too
DeadFish37 the pi function works only for x > 0. But the gamma function is defined for all real numbers except the negative integers. That's why gamma is more famous.
Zacharie Etienne Oh I'm very sorry! 😅 It's an error on my part.
Ask Euler
To me, the ∏ function always seemed more natural, because it hasn't got the extra "-1" in the exponent of t.
But they're exactly the same function, just shifted one unit horizontally, relative to one another.
I love these videos on interesting mathematical bits! Can you do one on Weierstrass functions?
So, what is the reason why the definition of the Gamma function is chosen in this weird way?
Whats 3! BRB time to do a wall of calculus to find the answer XD.
EDIT: its 6 apparently
actually its very easy to find gamma(n),n is natural...but i need to do this for pi and i have no idea how to XD
There will be a day when i will need this type of teacher
Love videos of Blackpenredpen
I looked for a video for this function just yesterday, perfect timing!
whats that box?
Nyc video!!
Are you going to do Beta function also?
& their relation , It turns out to be helpful in many cases!
Chaitanya Paranjape i can. But prob next week or so. Thank you.
blackpenredpen Yay!
make a playlist of all the alphabet functions! make sure to keep them in order.
Really, you are a great teacher and I'm excited to watch more videos about your lessons. Thanks for help
Nice video, It would be cool to see you make a video explaining the properties of the gamma function, overall great stuff.
Why there is a gamma function? Gamma is NOT the generalization (English is not my first language) for factorials. Genaralization for factorials is Pi function. And pi function includes x, but gamma includes x-1 and gamma(x)=(x MINUS ONE)!. Pi is pretty useful.
In this video you show that the pi/gamma family of functions are able to extend the factorial function to the reals. Could you prove that this family of functions is unique? ie no other function maintains the listed properties for the reals.
Great question!
iirc he needs to also add the condition of log-convexity
Why must the function be logarithmically convex? My guess is that the first two properties imply that the function will be log-convex, but idk.
Because otherwise it's not unique (in this case). Look up "Hadamard's gamma function" (it maintains the two properties of f(1) = 1 and f(n) = n * f(n-1) but it's not log-convex)
a convex function is the same as a function which is concave up. more specifically if you pick any two point on the function the connecting segment will be either on or above the graph. You may have checked for this calc class using the second derivative test. a function is logarithmically convex if the function log(f(x)) is convex.
en.wikipedia.org/wiki/Logarithmically_convex_function
Great Explanation! I had alot of fun watching the video. Thank you.
Supreme jacket CL0UT
LHopital's rule is overkill for these limits. Just use arguments using inequalities.
3:33
f(1)=1
f(x)=x*f(x-1)
uh i think that's exactly
what i used
for function factorial
-in javascript-
Hey ! You made it !!!! Do the integral of 1/1+sqrt(tanx) !
You make it seem easy!!! So brilliant :D
I finally understand the gamma function. thankyou!
Dahm, this is a whole new level of fascinating
Thanks for video
Very good explanation
So what's the point of the gamma function, anyway? The pi function seems like a much more natural extension of the factorial. But for some reason the version that's confusingly shifted over by 1 is the one that's always taught?
FourthDerivative
The gamma function pops up everywhere.
Okay, but still, why not use the Pi function in those cases instead? They're literally the same function, just shifted over by one, and Pi has the advantage that it has a more straightforward correspondence with the factorial over the integers. It's like the tau vs. pi debate, the baggage of historical notation just makes things unnecessarily complicated.
The only good use of gamma over pi is that the first "pole" or blowup of the gamma function is at the origin rather than 1. This makes some contour integration in the complex plane a little bit simpler, but other than that it ruins all the formulae. I wish the mathematical community had stuck with pi.
Well, sometimes things appear so often they need another name. In chemistry/physics we use the Dirac constant all the time, even though it can be expressed in terms of the Planck constant.
ħ=h/2π. Why? Because it pops up so much it just makes the notation cleaner.
Best Content on Whole TH-cam!!
Great... I am always appreciating to you.
Very nice intro to the factorial in terms of the Pi function.
Then why do we need a Gamma function at all?
Can you please explain that?
As I understand, the Gamma function ALSO generalizes the factorial idea to ALL real (or complex) nos. Then why do we need the Pi function at all?
Honestly, I am seeing the Pi function for the first time.
Would be grateful if you can share a link about the Pi function and it's application
:O This was an amazing video!
Can you do a video on the Riemann Zeta function (and maybe the Riemann Hypothesis and the infinite sum of 1/n^2 =pi^2/6)? I'm curious as to how Riemann was able to come up with the integral.
Thank you so much for your brilliantly clear and enthusiastically explained videos! I have a question though: what's the point of having both the Pi and Gamma function? Surely having only one also does the job of the other? What do they add to each other that the other doesn't have?
Two things I’m curios about, so if someone could explain it’d be much appreciated:
1. How did gauss come up with the function? It seems very arbitrary for the exponential integral to be used as an extension of the factorial function and doesn’t seem to derived very directly, except by comparison of results.
2. Why is the Pi function called the pi function? It doesn’t have pi in it unless you use eulers identity.
3. Where do the imaginary results come in?
because people are too uncreative to name a function so they just plug in random greek letters. nothing to do with the 3.1415926 constant.
the capital Pi is for product, which the factorial basically is[at least for integers].
thats not 2 things its 3
Great video!
Can we get the Π (or Gama) function(s) from the initial equation, or is just an happy accident, i.e but studying this integral we figured out is had the same property as factoreo?
We can. I think it was weierstrass that extracted this integral from euler's infinite sum, but i could be mistaken.
That is really a good video. I also learnt how to do integration by parts quickly aside from the main content
You need at least one extra property to continuing the factorial, since now Π+sin(πx) is also a solution. You could introduce the property ln(Π) is convex
It's even worse as pi(x)+sin(pi*a*x) is a solution for all integer values of a.
14:33 best part 😊
0! can be found using the two conditions itself.
since f(1)=1 and f(n)=n•f(n-1)
From the second condition , if we put n=1 , we get f(1)=1•f(1-1) => 1=f(0) => 0!=1
Regarding Pi being "cooler" than Gamma: that depends. The functional equation for the Riemann zeta function, expressed in the form Z(1-s) = Z(s) where Z(s) = pi^{-s/2}Gamma(s/2)zeta(s), would be way less cool-looking if we had to use Pi instead of Gamma. Ultimately, this is connected with the fact that when integrating over the multiplicative group (0, infinity), it's usually way cooler to use the Haar measure dt/t than dt, and this offset by a factor of t explains why it's often advantageous to use Gamma instead.
Love your content. Keep it up 💯
This video is fantastic. Thank you.
+ bprp - I understand the "pin" has already been won; I don't see it right now (haven't searched through these comments enough), but I'll weigh in anyway, because I suspect no one has (yet) invoked the following method.
The infinite product at the end of your video can be written
P = ∏ᵢ₌₁ºº (1 + aᵢ ), where aᵢ = -1/(2i)
Now such an infinite product converges or diverges, as the infinite series, ∑aᵢ , does. But that series is harmonic and is well known to diverge (to -∞). Therefore, the infinite product diverges to 0.
[An infinite product is said to diverge if its limit is either infinite or 0; in other words, if its log(absolute) goes to ±∞.]
Fred
Please make more videos like that, about more complicated maths!
e^t never dies.....!
The best video of the year :D
It is too crazyyyyyy!!! Loved it!!!
so why even bother with learning the regular definition of the factorial when this seems to be the "better" way? has the pi function already replaced it or is there still a problem and if so what is it?
coldmash Why bother learning the arithmetic definition of exponentiation when one could just learn the Taylor expansion of it and then already have this be well-defined for all complex numbers?
I contest that there are an infinity of different functions that pass through the points (1,1),(2,2),(3,6) etc, it's just that you have shown the simplest such function. For instance, the functions could be sinusoidal, but multiplied by the pi function. That would work. I understand why people have hit upon the pi function, after all, it's simple to work with, but there ARE other solutions
I have not found any concrete evaluation of gamma(1/5) anywhere online, not even approximations. the simplified integral diverges and I can't find a close enough series to find what values its riemann sums converge towards
It would be cool if you could cover that in a future video
It was a useful and enjoyable lesson for me. Thank you
Good use of color.
loveee that supreme sweater man!
What is this method for integral by parts? Do you have some video about that?
Ricardo O. It’s just the regular definition of integration by parts. Integral(f(x)g(x)dx) = [F(x)g(x)] -integral(F(x)g’(x)dx)
Where f(x), g(x) are functions of x and F(x) is the primitive function of f(x) and g’(x) is the derivative of g(x). He is just using that “box method” as an easy way to remember how to assemble the right hand side.
He calls it the DI method. He has a video explaining it
Raphael Antunes thanks
Awesome mate , as always!
I have a question, sorry if I find misinformed but I learned to integrate by parts using another method. Int u*dv = uv - int{v*du}
And this DI method I’ve never seen, specially when you used a third column to give the answer to the integral . Would you mind sharing where I could learn it?
Always nice to get to know other ways.
Thanks a lot e keep it up my friend
I second that! 😉
00:40 Also for `n=0` ;J And if you find empty products fishy, as some people do (not me; for me, an empty product is just "take the unit and don't transform it in any way, so it stays unchanged"), you can derive `0! = 1` from the second property you mentioned in 3:14:
Since `n! = n·(n-1)!`, we can divide both sides by `n` to get:
n! / n = (n-1)!
We can easily calculate `n! = 1`, no problem with that. But if `n=1`, then the right-hand side is `(n-1)! = (1-1)! = 0!`, isn't it? :>
And it has to be equal to the right-hand side, which is `n! / n = 1! / 1 = 1/1 = 1` :> So it must be that `0! = 1` ;)
It also shows why the factorials of negative integers are undefined: choosing `n=0` we find that there's 0 in the denominator on the right-hand side, so although we can calculate `0!` now, division by 0 turns out to be problematic :q This will also render it undefined for all other negative integers, because of `n!` in the numerator being undefined for their plus-ones.
04:15 Yay! Finally someone else who knows the `Π` function! :> I always wondered why do people prefer `Γ` even if it's more cumbersome than `Π` due to that "shift by one" :P I smell a rat here, because when you check out the history, it didn't even started out as the integral we know today. Euler introduced it (when studying Goldbach series - a sum of subsequent factorials) in a form of an infinite product, which is much more clear, because it obviously connects with the product in the factorial. The integral form is a later invention which only muddies the issue and makes it undefined for negative arguments :P (contrary to the infinite product, which is defined for all complex-valued arguments all fine :P ). Euler came up with it when he noticed that his infinite product reduces to the Wallis's product for `π/2`, and once he saw that `π` is involved, he immediately thought about circles, so he switched to quadratures of the circle expressed with integrals. But now we're being taught backwards, putting the cart before the horse :P Not only that, but no one explains us where do these fancy integrals come from and how did Euler and Gauss figure them out :P And we have to deal with that incomplete definition with integrals that don't work for negative arguments and "restore the order" by some pesky ways of analytic continuation :P Why, I ask?! But no one answers...
15:27 Indeed :> It's like with `π` and `τ` :q
16:19 That look ;D Like "Seriously, guys?" :q
16:45 Mah maaan ;J
Great video
"As always, that's it" ahahah. Good video, thanks
what a smart guy!
So if pi (0)= pi (1)= 1 then the pi function have a minimum between 0 and 1 right? What are the minimum coordinates?
Yes, it does indeed have a minimum between 0 and 1. The minimus is at (0.4616, 0.8856).
Close to (e^2)/16, but not quite. How is 0.4616 actually derived?
I just used Desmos to graph x!, and looked at the min co-ordinates. You can try to derive it yourself, I just don't think it will be easy.
Use the derivative and set it equal to 0.
+ simon rothman: Or use Grapher in MacOSX; or Wolfram Alpha (math calculation free website extraordinaire).
Not sure what that number is, mathematically. But shifted by one unit; ∏(x) = Γ(x+1); it's a zero of the digamma function, which is the derivative of Γ(z).
But it (x ≈ 0.4616) almost satisfies (x+1)² = 1/x; i.e., x³ + 2x² + x - 1 = 0.
Thank you for doing this video.
Maybe x! should be defined as a family of functions rather than a function itself.
x! = {
x when x = 1
x(x - 1) when x = 2
x(x - 1)(x - 2) when x = 3
x(x - 1)(x - 2)(x - 3) when x = 4
. . .
}
Each of them is its own polynomial function that can be used to compute a unique factorial.
We're not playing hangman here😂😂😂😂
Blackpenredpenbluepen.
12:20. People who are familiar with big O notation could probably skip over this lhopital's rule because O(constant^x) > O(x^constant). This makes all instances of lim(x^constant / constant^x) as x -> inf return 0.
• 14:24
• press C
• discover a new 'limit' .. namely that of automated speech recognition.
If n is not an integer you can't just take n'th derivative of t^n in lim t-> inf (-t^n/e^t)
indomit here (t^alpha)' = alpha*t^(alpha-1) even if alpha is real. So at one point the exponent it to will be negative so the numerator will approach 0.
yes, I know it. I just think this remark should be in the video.
Thank you it is really helpful
There are other reasons why 0! = 1 is the "right choice". Almost every formula that has factorials reduces down to the "right thing" when you make that definition. So it's good that Pi agrees with that.
For starters, consider "n choose m" when m = n. How many ways are there to pick 5 objects from a collection of 5 things? Well, you clearly just take the set itself. And n choose 0. How many ways are there to pick nothing? Well, there's only one empty set. n!/(n! 0!) = 1, so we're good.
I like the kokoro odoru look
15:08 you have prooved by induction 😍
Interesting extending factorial
Could you bring the caméra a little closer please sir the teacher
Thank you teacher 🙏
came to learn about gamma function... It is a video about Pi function + gamma function equation no explanation
This is awesome!!
This is soooo helpful, thank you so much! 😈 !
This is beautiful
noone can explain mathematicaly why is 0!=1, its cool! explain was excelent!
You have so much content 😍
14:39 made me laugh unreasonably hard
thanks a lot! so the factorial is defined in positive integers and pi funciton in real numbers??
Man, i actually understand this at some point
What tells us the Pi function is the only one that meets those conditions?
There are others. Example: you take pi func and double it everywhere except integers
@@dudono1744 "Those conditions" include continuity and differentiability. Your function is none of those.