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What is the Gamma Function?
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- Опубликовано: 9 окт 2017
- Question 7 from Tom Rocks Maths and I Love Mathematics - answering the questions sent in and voted for by YOU. This time we've got probability distributions, complex analysis and of course Pi (because it appears everywhere)... I give you the Gamma Function.
Full playlist: • How many ping-pong bal...
Q1: What is the probability I have the same PIN as someone else?
Q2: How long would it take to sink to the bottom of the ocean?
Q3: What is the gravitational field of a hollow Earth?
Q4: What is the best way to win at the board game Monopoly?
Q5: What are the most basic Mathematical Axioms?
Q6: How does Modular Arithmetic work?
Q7: What is the Gamma Function?
Q8: How many ping-pong balls would it take to lift the Titanic from the ocean floor?
Q9: What is the graph of x^x?
Q10: How can you show geometrically that Pi is between 3 and 4?
Produced by Dr Tom Crawford at the University of Oxford.
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There are still another 9 videos for you to enjoy in the series - find them all here: ruclips.net/video/by8Mf6Lm5I8/видео.html
I met you watching a Numberphile video and fell in love with the way you communicate. And today I found out your maths channel. Thanks a lot for making my day!
Welcome aboard!
I thoroughly enjoyed this video. I not only learnt about the gamma function but also about a useful approach to solving difficult integrals, and I was entertained and inspired by your energy and enthusiasm. Thank you!
You're welcome
Your explanation is as much as a mission about fantastic discoveries and joy of maths as understanding something as a very practical tool to manage a lot of phenomenons in real life.
You look like a popstar man
@Deka same here :D
You're too kind
@Deka I don't know why but me too!!
@@VANSHSHARMA-iu8eo you know why
@@Joe-bb4yi no , please tell me.
Frikking incredible! I've tried & tried to understand the gamma function and you've made it crystal clear in under 12 minutes. Thank you, Thank you, Thank you, !!! I'll be watchin' you, boy.
Thanks Niawen - glad I could help.
Tom!, u r not just understanding mathematics.. U r talented in delevering the msg.. Thank you so much.. I need from you more videos about Taylor expansion, Lagrange multipliers, and more of special functions.. We know the names but it's all empty and easy to forget the second after the exam!,
And again.. Thanks in advance
Thanks Weaam!!
Though I'm a late viewer of your teaching, but it pushes me up to rethink mathematical stuff and helps to study various branches of Maths as well. An emphatic love and respect by my side from India.
Thanks Anirban!
You just make my life easier everytime i tune in
9:10 Sigma squared is the variance. Sigma is the standard deviation.
Your manner of presentation is very cool. After watching your video I have finally understood what is it. Thank you very much 👍
Awesome - glad it helped :)
Thanks man, I didn't understand a thing when my lecturer taught me and suddenly boom! quiz tomorrow. This video literally save my ass
Happy to help :)
Amazing video. I hope you continue to make videos like this!
Thanks Hans - and yes that's the plan!
I had always heard of the gamma function but never knew what it was. Then one day I decided to plot factorial function on desmos and that crazy graph with monster asymptotes appeared lol so I had to come see with this was about, thanks for great video
Haha love the story - and you're very welcome.
I'm heard for the exact same reason
Great video tom. I greatly enjoy your videos. Thanks. Keep up the awesome.
You are simply amazing, man. Immediate subscribe.
Amazing clarity! Thank you!
You're very welcome.
Rockstar mathematician!!
Explanation is good✌️
Wow man love from India 🇮🇳🙏🇮🇳
Thanks!!
Clear, complete and so well explained. Thank you. :-)
You're very welcome Andy.
Hey, your explanation was amazing man. Keep it up. Love from India.
Thanks Apurv!
Kaisa gya advance
Great man, just great
I have been trying to understand how the integration by parts was evaluated for the polynomial and exponential function in multiplication there in gamma function.
But what's the name of the rule you talked about at 5:16
Thanks you have been very helpful 😊
Great Professor ... Great. Especially the last part (distribution function).
Thank you so much.
Professor, please let me tell you something (maybe it's not important for you, but for me ...) :
Most of the time I see, you're kneeling in front of chalkboard! It means a lot to me. Because and In my opinion, it shows how much you love Math and your work (actually your Hobby!). And also it shows, you're such modest person.
Professor, you are such a real and great teacher.
Thank you
Thank you man ! It is clear now. Love from France.
Glad it helped Juliette!
This video is incredibly helpful!! Thank you!
you're very welcome :)
Great video, thanks. Just a typo: In the bottom right graph from 3:20 to 7:30, the y-axis should be the gamma function and the x-axis should be x, the real argument of the gamma function.
I admire your work. Thank you.
i jst had to subscribe man you solved all my problems of gamma
Awesome - thanks Phumudzo.
I thoroughly enjoyed this.
I enjoyed.Please explain Lars V.Ahlfors proof of gamma(1/2)=root pi
I'm glad to hear that Che!
Quite nicely explained....love from India....keep up the good job👍👍👍👍
Thanks Tanmoy!
Good job man!
I hope that you will became a successful man in math.
Just watch this impressive Math channel ruclips.net/channel/UCZDkxpcvd-T1uR65Feuj5Yg
I must say thank you
I have these gamma beta functions in mathamatical physics which kind of made me confuced
thank you for your explanation
you actually helped me to start studying
Awesome - happy to help :)
Nice video
Great video sir 👏
Thanks 👍
It's amazing explanation sir!!
Thanks Rekha.
The Gamma Function is Punkatortially Glamorous in this Tutorial.
Nice - thanks Max.
U should become a math professor. You are way better at explaining ideas than so many math teachers i've had.
avenging209 that’s exactly what I do!
lmao he's a tutor at St Hugh's College, University of Oxford..........
You are such a great teacher.
INCREDIBLE ACCENT. +100% CLARITY.
Great video Tom! Appreciate it! you explained the process of computation for the gamma function well. Why not discuss its significance a little bit? Like what it simplifies or reveal?
This channel is so underrated
his intro killed everything
Great video! It was fun to listen to. But where were the complex numbers?
Your explanation is the best, in spite my English being very bad, i understand it.
Great to hear - thanks!
Love your way of explaining can I ask a favor? can you do a video on beta function?? I am currently studying it in my mathematics class and I found your method of explaining is
easy to understand so thank you on the fun and informative video and can't wait for your reply
Thanks for the suggestion Muhammad, I'll add it to my list :)
Very good explanation
Thanks Zeba!
Hi Tom, thanks for your clear math video sharing. I have a question here: around 5:10, you mention about a magnitude limit rule. Could you tell me the terminology of this rule? I'm not a native speaker and in many cases it's hard for me to link the English term to my language. Thanks a lot!
Thanks for the question - I think this page does a good job of explaining the idea calculus.seas.upenn.edu/?n=Main.OrdersOfGrowth
Sir you cleared my concept 😀
Awesome :)
Great ! Thank you !
You're welcome Doktor Klaus!
thanks!
thank u so much sir for this video and i can safely say that i draw many analogies from bodybuilding to the branch of mathematics ..just like in body building all parts are trained and sculpted so do here every piece of mathematics must be understood well before constructing the bigger picture
glad it was helpful!
This is how you break stereotypes. #RocksMath
Thanks Aman - I'm trying my best!
I have a question, or rather an idea to propose: wouldn't it be a lot easier to solve the integral for gamma of half with polar substitution rather than going all in with probability distribution?
If I understand correctly, root pi comes from the fact that the area under the bell curve = root 2pi . . . what's amazes me is that both e and pi are jointly involved in what can be considered the most powerful force in the universe - the law of averages...
Yes exactly. It's all coming from the Normal Distribution/Gaussian Curve. More info here: ruclips.net/video/xp3J_uSYtD8/видео.html
From all the math guys i saw you are the most "not math guy" looking guy and thats not a compliment.
Great explanation. Thank you!
You're very welcome Bekoe.
Thank you
You're welcome :)
I was playing around with a graph function f(x) = x! and f(a) = e^a and that they grow at the same rate eventually
Please do a video on differentiation under the integral sign
will add it to my list - thanks for the suggestion!
i have a question, at 3:11 you mentioned that this graph shows the full extent of the gamma function but shouldn't that graph be three dimensional(rather than the 2 dimensions shown), you have a Re(z) and Im(z) axis but where is the F(Re(z)+Im(z)) axis ?
fau s I think the graph he shows here is a graph of gamma(x), so only for exclusively real numbers. I think he made a mistake here; the axes should be defined as x and y, not Re and Im. en.wikipedia.org/wiki/Gamma_function
Good Boy! It is like that!
I'm pretty sure its a graph of the domain of gamma function on the Re and lm axes
Good spot Fau - the graph is indeed only for the real part of the function. The x-axis is the real part and the y-axis the value of the Gamma Function. My bad.
I very much appreciate, if you could do a derivation of the normal distribution, that would be great.
I discuss it here in fact: ruclips.net/video/xp3J_uSYtD8/видео.html
Great video! I'd just like to point out that the denominator in phi(x) should contain sigma, not sigma squared
Thanks man.
You're very welcome Tawfique.
could someone explain how this could be used to explain the forgotten index of the Fibonacci cube Gamma(n).
3:14 "Integral can be extended on the left side of the complex plane except for negative integers". I did not understand this part. Can anyone explain?
When I look at this graph, I think I can see line plotted in negative quadrant.
At 3:11 how do you draw the graph for the Gama function?
Hi Karthik, the graph I've drawn is the real part of the function (sorry I mis-labelled the axes) as the full graph gives you a two-dimensional surface. There's a nice plotting tool on Wolfram Mathworld which lets you play around with different values, I recommend trying it out: mathworld.wolfram.com/GammaFunction.html
This is great stuff!!!!!!! I spent so much time learning Gamma from video lectures... I continue to seek intuition on the integral form.... I think the gamma function is actually fits best on a 3D plot... on a 2D plot its mostly depicted as a Real vs Real 2D map.. in other words, I think the 2D map is wrong.... and to make it right we gotta replace Imaginary with Real, and imagine the Imaginary dimension extending into and out of the blackboard! Blackboads are soooo outdated!
oh, that guy.
Great👏
Awesome!
Thanks :)
I wish I could understand what it means to have an integral from a to b. I want to know how to calculate 0.25! but I don't understand integrals and it will take ages to understand it by just researching it.
so useful to me, everything's great! except the side camera's resolution...
Thanks Li - fortunately I now have a new camera so my latest videos should be much clearer :)
can any one recommend a text book that I can study this
Nice video! :) at 9:13 don't you mean sigma squared should be variance and standard dev should just be sigma?
yes sigma squared is the variance
Exactly
Where are you from??
Yes sigma squared is indeed the variance, and sigma is the standard deviation. Apologies if I mis-spoke.
Can anyone tell me where this Gamma function comes from? I've read about it, studied with Z being a complex variable and read the history but no one can tell me where this function happened or in what circumstances these mathematicians found it.
Please if anyone. I'm curious.
Great example and explanation… however, you didn’t happen to mention what the Gamma Function is actually god for or why it’s used. It leaves me still curious about the application of the function.
Cheers
I have a question that I hope you can answer. During my studies of the gamma function, I came across a relation known as the "Stirling's Factorial Approximation." The equation is commonly used to calculate the value of gamma(p) when p is very large. Anyway, when choosing a value of p like 450,000 and substituted that into the equation, I always obtained a value of zero, even though the gamma value at p=450,000 exists. I later discovered that in the equation there was an exponential factor raised to the power of -p. We know that exp(-p) =0 when p>>0. How can the equation have got this wrong? Or is there something I am missing. Big fan of your channel! Thanks in advance :)
Hi Rafiq, thanks for your question. The value of gamma at p=450,000 is actually really large (I got the answer 10^(10^6.370792540767548)). You're right that the exponential function would make such a term small, but if you look carefully at the integral definition, the exponential is actually unchanged no matter which value of gamma that you are calculating. It is an exp(-x) term, where z is the variable in the gamma function. This is why the gamma function will tend to infinity as the input variable tends to infinity.
Thanks for the swift response. I understand what you're trying to say. The integral definition of the gamma function does indeed have an exponential term that is independent of the gamma function variable "z." But if you recall, my question was concerning the Sterling's Approximation Formula. Here is the equation, perhaps you can calculate the value differently:
gamma(z+1)=z!=(sqrt[2*pi*z])*(z^z)*(exp[-z])
Apologies I overlooked the fact that you were referring to the approximation, rather than the integral definition. Using the approximation formula as you have given above, the answer will still be very large. Again, I tried inputting your value of 450,000 into the formula on wolfram alpha and obtained 10^(10^6.370793586177315), which is a very close approximation to the answer I obtained above for the exact gamma function. The reason the function continues to increase is due to the z^z term. You are correct that the exponential term exp(-z) will quickly descend to zero for large z, however, the z^z term is also an exponential function. In fact, it will dominate the exponential term for any z>e. To see this, substitute in the value z=e. The z^z term then cancels exactly with the exp(-z) term, leaving only (sqrt[2*pi*z]). Furthermore, if you rearrange the formula by grouping the exponential terms together you have:
gamma(z+1)=z!=(sqrt[2*pi*z])*(z/e)^z
Now, hopefully it is clearer that as z tends to infinity, and in particular for z>e, the exponential term to the power z will increase very quickly towards infinity also. Hope that clears it up!
Got it. Thank you very much for the explanation. Sorry if I made you tired with my question. :)
No problem!
The value of √(1/2)= ? a)1/2 b)√π c)√π/2 d)1 .I was asked this question in a test. We don't know the answer,so we asked our teacher ,he said the answer by looking the gamma function in the book where Gamma (1/2)=√π was in the book. I have a doubt that is gamma and root are same?
I had to remind myself that when doing gamma(1/2), its not (1/2)! But then to work out what (1/2)! is, you need to calculate gamma(3/2) which is equivalent to 1/2 x gamma(1/2). That is 1/2 x root pi.
Just watch this impressive Math channel ruclips.net/channel/UCZDkxpcvd-T1uR65Feuj5Yg
Genius
Thanks Hardik.
Is it arithmetic progression
you explained gamma function for positive integer(n)...please explain for negtive integers(-n)..
The same integral formula will work for any value of n.
Thank you , could you cover beta-gamma relationship?
Exponential Integral Gamma function??
Pretty much yes.
That’s interesting. N-1!
Hi your content is so good ... informative and easily understanding... But just improve the quality of the camera and only one the front view is sufficient don't move the camera view it diverts the focus... Please ...
You make esay gamma function for me
thanks
Happy to help Suneel :)
Is there way to find a number if you are given only its factorial with some inverse gamma function?? Its really bothering me.. i need answers!!!
I think this might be what you're looking for: mathoverflow.net/questions/12828/inverse-gamma-function
@@TomRocksMaths Perfect .. thanks!
I was upset because the cheetah t-shirt was blocking the equations, which should have been tattooed on your body or silk-screened on the cheetah t-shirt... why is there not a Gamma day?! That would be Factorial!... just kidding I'm not upset, nice video, very concise, which means I didn't get lost in the details!
I’m struggling with the u=√x how is it dx=x^1/2(2) I’m getting x^-1/2 (2)
When you divide both sides with what was multiple to du the half becomes 2 and the negative sign changes to positive
03:03 Actually this graph shows the gamma function for real values only (you say almost any value for all complex numbers at 03:10)!
Yes my apologies!
Love from India😍😍😍
You said in factorial you multiply all whole numbers less or equal but it must be natural numbers
yes all positive whole numbers
🎉🎉🎉
wasn't it root pie over 2 though??
I'm not sure exactly which part you mean, but the 2 that appears inside the integral after the substitution cancels out with the 1/2 factor to just give root pi.
Music Incorporate I heard it was sqrt(pi)/2 as well.
No, the factorial of 1/2 is root pi over two.
The gamma value of 1/2 is root pi
@@jainilshah6712 dose that mean that the factorial of -1/2 is root pi?
Hi
If you become our teacher ......Girls will attend your every class
The vocabulary you use when describing maths makes it sound very exciting.
I understand the basics of the gamma and the basics of factorials now. Thank you! I loved this. Something to add to my 10 year old brain...
What's wrong with being a nerdboy eh?
@@egeerkut9602 hehe bence adamın kastetmek istediği yaşının 10 olduğu değil mütevazilik yapmaya çalışmış ama olmayadabilir xd
Nobody cares how old you are
Sounds like a bunch of people in the comments are jealous.
love from india
The graph at 3:03 does not make any sense, because the axis are labelled wrongly.
I guess you wanted to show us the Gamma function restricted to the real line.
Just watch this impressive Math channel ruclips.net/channel/UCZDkxpcvd-T1uR65Feuj5Yg