In this video, I showed how to obtain then gamma function by simple integration and repeated application of Leibniz's Integral Rule Buy the t-shirt here shorturl.at/HNUX1
Have just recently started to watch your videos and your enthusiasm for maths is infectious! I just wish my high school teachers and professors of the 70s had that inspirational spark!
Lol and Me revisiting Gamma functions this way at the heart of the matter! 13 years have passed and nobody teaches the derivation of CRUCIAL functions like these to engineers probably because they thought it’s irrelevant but the point here is when you do a derivation you open up new doors to possibilities and your Ebbinghaus curve would be smooth as ever (you’d remember things even better!)
Hello Mr Newton. This is a great video And I really enjoyed it. I have never seen it done this way before, and I have an MSc in pure maths. It's so clear and simple. 📳📴✅
I met the Gamma function about three days ago in the Fermi-Dirac integrals and somehow, without searching for Math tutorials, I bumped into this. How cool?
I LOVE your videos. There's so much dedication, GREAT explanations, POWERFULLY INTERESTING math ideas. Easily one of my favourite math channels, if not my favourite one. Keep doing as great as you always do 8)
I have Totally forgotten Calculus I can’t follow this Actually. I need to Start at Trigonometry going to Pre-calculus I am at a Algebra 2 level or College Algebra level with some Trigonometry knowledge.You an a extremely intelligent person and one hell of a teacher You have a passion for it I love your attitude I am 66 I will be auditing Trigonometry at my local college this fall Then taking Calculus 1 I have a young student who I am tutoring in Pre-Algebra He wants me to to be able to help him out in Trigonometry and Pre-Calculus Actually He is Algebra 1 Ready He was totally failing math The light has been turned on And all cylinders are firing He is a freshman in High School He can pass Pre- Algebra now They are going to let him test out so he can Take Algebra 1 his sophomore year will be teaching him Algebra 2 now and all throughout the summer He wants to test out of Algebra 1 This fall He wants to take Algebra 2 and Geometry Junior year Trigonometry Senior year Pre- Calculus …Soo By the end of next year He will have the same math knowledge as I have right now So yeah I will be auditing math courses this fall …Yes never stop learning and it is never to old to learn It my case I forgot I did it once before I can certainly do it again And I can’t let him pass me up
So are you conceding that you did the "illegal" things in the last video? Because, yes, you did. I'm glad you mad this response (and it's cool you have responded so quickly)
The discussion around the backward factorial development and the gamma function have been enlightening to me. Finally this stuff makes sense to me. Well: People like me, who just got an engineering level math education, get the equivalent of a lecture with this videos. Keep it going! :D And concerning the content of this juggling here: That looks like a good example for the justification of mathematicians playing around with things they have just discovered to stumble by accident upon completely new stuff that blows the mind when finalized :D When i looked through wikipedia articles after the previous video on factorials, i threw the towel when it came to the deduction of the Gamma function, but with this explanation here it is perfectly fitting in to my pre-knowledge. Thanks!
Hi, Awesome! I've been trying to find a way to derive this for a long time, and you just did it! Thanks a lot! Something else : I've been working on a way to write the factorial function as a polynomial series + a rational fraction series for a while. Say : x! = a_0 + a_1 x + a_2 x^2 + ... + b_1 / (x+1) + b_2 / (x+2) + b_3 / (x+3) + ... For now I have demonstrated that the poles (the negative integers) are single, which is quite easy. I then tried to write relationships between the coefficients by applying the formula: (x+1)! = (x+1) x! and by indentifying coefficients. But it's a bit difficult, you quickly get complicated formulas and you are kind of sailing backwards. By taking x=0 or x=1 you get some simple formulas, but that's all. Do you know a better way to do that?
@@bigfgreatsword That is why we should redefine it to be ℼ = 6.28... This way we have ℼ radians in a circle. (Oh, and for nerds/geeks we have the fomula exp(ℼi) = 1.) Bottom line: 𝜏 is just such an ugly symbol for the job!
when you assumed that the area was half when you took half the bounds, you should’ve proved, or at least mentioned in passing, that it was because the function was symmetric
I really like this presentation. I suspect it lends itself to calculating the Gaussian integral without a complicated Feynman trick in the exponent. I typically derive the factorial by trying to find the Laplace transform of $t^n$, but that's not as parsimonious as this approach.
I have a related question. Is the intuition behind it just that partial derivative with respect to t and the integral of x are constant relative to each other? I'm not sure if the proof goes deeper or if the proof's complexity is largely rigor. Full disclosure I haven't looked into it much yet
Great video as always, but I'm confused why we put t=1 like are we allowed to assume this or just to make things easier , and if so why not other number like 2,3,4 etc... . And again thank you so much for thus great channel ❤
Iiuc the integral with t in it is more general than the gamma function. In other words, the gamma function is a specific instance of it. Prime Newtons showed us how to prove that the more general integral was equal to factorial over t^Z, and then showed that replacing t with 1 gives us the gamma function.
The idea is that this is a general explicit definition of the gamma function, which works for all real t. Setting t = 1 just makes for a simpler expression.
Very cool video but how does the Gaussian integral fit in to this? Doesn't changing x2 to tx change the nature of it, especially given that t isn't a function of x?
Phew! Truly a thing of beauty! But how do you think that the discoverer of the Gamma function started the derivation with the integral (from 0 to infinity) of e^(-tx) dx? Do you think that he/she already knew the "destination" and reverse-engineered to get there? For example, noticed that if you keep differentiating e^(-x) dx you'd get the form of a factorial as the multiplier?
I have a question regarding the step taken at 1:39. I can clearly see it works here, but does it *always* work? In other words if given some f(x): R -> R and real number L such that Int{-inf to inf} f(x) dx = L, is it always true that Int{0 to inf} f(x) dx = L/2?
@@ingiford175 Ah, yeah, that makes a lot of sense. It hadn't occurred to me until you said so that e^(x^2) is an even function because, for whatever reason, it doesn't really "feel" even to me.
@@naturallyinterested7569 Yeah, I agree. Because if you look at the last expression, you could just come back and resubstitute n = z -1 and it's just a simple expression for n! There must be something else here.
Oh. That's the only way you can enter the input directly as the argument of the function. Otherwise, you'd be writing Gamma(x-1) or Gamma (x+1) and not Gamma(x)
@@PrimeNewtons But that's exactly what I mean, without this shift in the definition of Gamma(x), for which I know no reason, we would have Gamma(n) = n! What I don't know is for what reason (other than to annoy me ;) is that shift there?
The factorial of a negative number is UNDEFINED or INFINITY. So, gamma of ZERO is infinity. And So the logarithmic value of negative number is imaginary.
Wait, 0! Isn't supposed to work? The number of arrangements of a size 0 ordered set? You have only one possibility: take none (which is taking all), thus 0!=1
I think that's sort of the point of the video. If you define factorial simply in terms of set theory (permutation of n distinct objects) then size 0 set doesn't make sense. But it's observed that the repeated differentiation of that integral can ALSO be a definition of 'factorial'. And in that context, we have a different way to calculate n!. Using this new method definition, it DOES have a value for 0! and 'can be shown....' to have a value of 1.
In math, sometimes things have different meanings depending on context. Like 'parallel lines' in flat plane geometry never meet. But in non-Euclidian, 'parallel lines' can mean something different and in that context they can. Maths.... what can I say?
@@mikefochtman7164 except that a set of size zero makes perfect sense, it is the empty set, which has precisely one permutation, so my confusion is why some would claim that 0! is undefined in the classic sense
@@ProactiveYellow But he didn't base his derivation on the interpretation that it is the number of permutations. He used the function n! = n(n-1)(n-2)...3*2*1 and then tried to slip in a 0 for the last term. As was pointed out in the last video, you can't do that because the factors in the function necessarily terminate at 1. If he would have used set theory, it would have been a different argument.
@flowingafterglow629 But then it would be an _empty product,_ that is a product of an empty list of factors, which by convention is equal to the neutral element of multiplication, that is 1. In the same manner, like an _empty sum_ is equal to the neutral element of addition, that is 0. So it makes perfect sense.
