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. 📳📴✅
Another, more well-known and arguably more general, way of approaching the Gamma function (defined for positive real numbers x) is doing it the other way around. You start with the Euler integral Γ(x) = int_0^inf{ t^(x-1)exp(-t)dt } and try to evaluate it doing integration by parts. This is very easy, a recursion will show up, and you will soon arrive at the functional equation for the Gamma function: Γ(x+1) = xΓ(x) with Γ(1) = 1. When you now restrict x to be a natural number n, you will get Γ(n+1) = n! or Γ(n) = (n-1)! (which is obvious but can be proved rigorously by mathematical induction using the functional equation in the induction step).
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)
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)
I admire you sincere smile and joy and the way how you explain things. You made my day, Mr. Professor! And by the way, could you share where you graduated from (higher education). Thank you in advance.
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!
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.
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?
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
There's nothing wrong with 0! = 1. There are multiple ways of defining the factorial function (for natural numbers including zero), but they all agree on this value. For example, in one of the most popular definitions, when defined as the product of the first n positive integers, with n=0 we get an empty product which is defined as 1 (the neutral element of multiplication) by convention.
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?
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
@@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!
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?
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
@@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?
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.
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.
The best explanation of the gamma function I've seen in my 70 years.
Thank you
Why can't I understand 🥲
@@thokozanimwale2109 watch again, maybe if its too advanced, come back to it later after understanding the base concepts required to understand it
me too..as an ex calculus private teacher ( 70 yo also)
@@thokozanimwale2109 study more!
You’re the coolest maths teacher ever 😊
❤
Fax
yes
"In a previous video, I was accused of performing illegal activities"
Best start to a math video 😂😂
Hey Prime Newtons, I must say that you have an amazing talent. I watched this video for 18 minutes without getting bored. That is rare for me.
same
Absolutely the clearest, easiest-to--understand derivation of Gamma that I've ever seen - and I was a math major.
Wow! Great lesson! I love your chalkboard penmanship! ❤ 😊
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!
I see the passion and pure interest in your eyes while you explain things. Thank you sir.
these are the only videos i can watch all the way through and never get bored
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!)
Prime Newtons.... you are Fantastic Teacher. Congratulations!
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. 📳📴✅
Glad you enjoyed it
Another, more well-known and arguably more general, way of approaching the Gamma function (defined for positive real numbers x) is doing it the other way around. You start with the Euler integral Γ(x) = int_0^inf{ t^(x-1)exp(-t)dt } and try to evaluate it doing integration by parts. This is very easy, a recursion will show up, and you will soon arrive at the functional equation for the Gamma function: Γ(x+1) = xΓ(x) with Γ(1) = 1. When you now restrict x to be a natural number n, you will get Γ(n+1) = n! or Γ(n) = (n-1)! (which is obvious but can be proved rigorously by mathematical induction using the functional equation in the induction step).
I really enjoy your lectures, your way of explaining is very cool 🌟❤️
INCREDIBLE VIDEO!!!! its insane how well you explained this... Thank you for this explanation!!!
Instant subscribe. Wonderful, keep on "tap tap tapping".
Very clear description and board work. One of the best.
I love his facial expressions and cool nature.
This guy's energy always amaze me!
when you said "beautiful" in the end of the deduction that's exactly the word I was thinking, I love this channel
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?
It's all fun & games until the Fermions show up.
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)
What can I say, just wow.
Never stop learning.
Thank you for addressing the issue in the last video.
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)
I admire you sincere smile and joy and the way how you explain things. You made my day, Mr. Professor!
And by the way, could you share where you graduated from (higher education). Thank you in advance.
Prime Newtons a god amongst men! brilliant thanks.
Man this is the best explanation I’ve ever seen
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!
Amazing video, I really love your enthusiasm
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.
Really you are an excellent teacher .Keep it up !!!
The most exciting Prime newtons video aside from the cover up method ngllll. This IS BRILLIANT
excellent work! Thank you for making this video!
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?
5 mins in, and I can't help but point out that you just derived the Laplace(1) =1/s
Oh my God I was thinking the same thing as soon as I saw the 1/t. This channel just gave us a 2 for 1 deal lol
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
There's nothing wrong with 0! = 1. There are multiple ways of defining the factorial function (for natural numbers including zero), but they all agree on this value. For example, in one of the most popular definitions, when defined as the product of the first n positive integers, with n=0 we get an empty product which is defined as 1 (the neutral element of multiplication) by convention.
Ты хороший, учитель жалко таких мало😢
Я наконец-то узнал как появилось гамма функция
This was the explanation i needed! Thanks!
Great! Looking forward to the next video in this series of videos.
Simply amazing. Congratulations!!!
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.
This is pure Diamond. Could you, please, bring some Integral Equations theories?
Great videos! Now I think we are ready for the LaPlace transform 😅
Awesomely done
my great respect 😀
Thank you sir for an amazing lesson
love your vidieo's
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?
Is it just me or is anyone else listening wondering if Bob Ross just started to present math here?
.. thank you for the nice video.
Love this guy
Always count on Prime Newtons! ❤🎉😊
Prime Newton =passion for Math.
I enjoy with your class
thank you teacher
Amazing 🎉🎉
great explaination liked and subbed
7:19 Sir could you kindly do a video proving the "Leibniz Integral Rule" ?
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
This is the rule of differentiating the image applied to L(1)
Yes L(t^{r}) = Γ(r+1)/s^{r+1}
Well done sir🎉🎉🎉🎉🎉🎉
Why was the Gamma function defined as 𝛤(z) = (z - 1)! and not simply 𝛤(z) = z! ?
We actually did that (see Π(z)) but then realized that we use (z-1)! more frequently so we just defined the gamma function as (z-1)!
The same reason why pi is 3.141... but tau is 6.283...
@@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!
@@bigfgreatsword yeah why not tau = 3.1415... and pi = 6.2831...
@@weo9473 convenience
Thanks!
Thank you for the support 🙏
Excellent
Great video
Wow love it ❤
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 think it the concept 'modern' concept of the gamma function first came up with writings between Euler and Goldbach
@@ingiford175 Whoever did it, was brilliant!
Amazing ❤
Elfantastico !! ✌
yo that was so cool!!! Thank you for this video I am actually in a state of math euphoria right now
Me again. How In the world did you get so incredibly good at this with a degree in culinary arts? Do you teach math somewhere?
Is it possible to calculate the integral of the gamma function?
Absolutely awesome!!!!!!!!!!!!!!!
How can we deal with this formula if we take another values for t?
4:57 There you assume that t>0 but what if t
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
x² is an even function and therfore symmetric
Don"t worry Master, u are a good Guy. The contraditory always be...
Can you end the video by showing the whole board, so i can take notes,......(Keep doing, you doing great)
I'll practice doing that. Thanks for the suggestion.
"Mammagamma" ~ The Alan Parsons Project
"In a previous video, I was accused of performing illegal activities"... We'll make a physicist of you yet!
15:00 the unexpected pause at the easiest step of the video
That happens a lot to me.
I still don't know why one does this shift from n to z. It looks like just an obfuscation. Does it bring any benefits?
n is generally perceived to be natural numbers. The gamma function takes a lot more than that.
@@PrimeNewtons Sorry, I don't mean the exchange of symbols, I mean the input shift by one.
@@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?
Is there any method to calculate the approximate value of gamma(1/3)?
Doesn't the limit depend of the sign of t? Because if t is negative then lim_{R \to +\infty} e^(-tR) = + \infty
It does. t > 0 had to be specified
@@micharijdes9867 Yeah! But youtube teachers tend to not be as rigorous
Gama 1/2= root pi...polar coordinate?
Hey! Thanks for your videos, friendo, keep up the work 😎
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?
It works because f(x) is an even function. If f(x) is an odd function then the original integral is 0 for any R, but the {0 to inf} can be anything
@@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.
please i need same explanation on beta function
💯
Elegant ✨!
"illegal" 😭😭😭 who are the police then
Euler
Yeah, right? 👍
Finally a black guy explaining math
and he starts video with WHAT words????
Why did you redo it at 6:10 ?
sorry, but I still don't understand how the gamma function is a function of z but the factorial is of z-1...
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.
No. Factorial of a negative INTEGER is undefined
Law abiding citizen newton yessir
sir please show how e is created
hello sir can you solve
lim n -> inf (1/n^2) * Sum[Sum[b^2-d^2,{d,3n,10cn}],{b,2n,5an}]
a train derives gamma function
Why does e^(1/Rt) become 0
It is because it says 1/(e^Rt), not e^(1/Rt) as I thought it did at first. In this case of course, e^Rt is very big and 1/e^Rt goes to 0
Why do I understand things when you explain it but otherwise, not so much?
Because you're a good learner.