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.
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!
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!
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
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.
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.
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.
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 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
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
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
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...
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?
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.
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 😊
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 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?
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!
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
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
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 ...
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 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 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!
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!
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.
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.
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
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!
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.
What is this complicated thing used for? Well, it is easy! All you need is this even more complicated thing that even fewer people understand! Just chuck that in and you are got to go fam
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.
There are still another 9 videos for you to enjoy in the series - find them all here: ruclips.net/video/by8Mf6Lm5I8/видео.html
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.
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
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!
Worth adding that Euler's reflection formula for the gamma function is really useful in understanding how it behaves for negative numbers.
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!!
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.
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 :)
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.
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.
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
Hey, your explanation was amazing man. Keep it up. Love from India.
Thanks Apurv!
Kaisa gya advance
Rockstar mathematician!!
Explanation is good✌️
Wow man love from India 🇮🇳🙏🇮🇳
Thanks!!
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 :)
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..........
Amazing video. I hope you continue to make videos like this!
Thanks Hans - and yes that's the plan!
Quite nicely explained....love from India....keep up the good job👍👍👍👍
Thanks Tanmoy!
Your explanation is the best, in spite my English being very bad, i understand it.
Great to hear - 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!
INCREDIBLE ACCENT. +100% CLARITY.
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
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!
Amazing clarity! Thank you!
You're very welcome.
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 :)
You are simply amazing, man. Immediate subscribe.
Great video tom. I greatly enjoy your videos. Thanks. Keep up the awesome.
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
Thank you man ! It is clear now. Love from France.
Glad it helped Juliette!
From all the math guys i saw you are the most "not math guy" looking guy and thats not a compliment.
The Gamma Function is Punkatortially Glamorous in this Tutorial.
Nice - thanks Max.
This channel is so underrated
Clear, complete and so well explained. Thank you. :-)
You're very welcome Andy.
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?
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
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 😊
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?
Very good explanation
Thanks Zeba!
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?
I admire your work. Thank you.
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!
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
his intro killed everything
Sir you cleared my concept 😀
Awesome :)
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
This video is incredibly helpful!! Thank you!
you're very welcome :)
This is how you break stereotypes. #RocksMath
Thanks Aman - I'm trying my best!
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
Thats pretty neat but what can the gamma function be used for? Can you derive Pi from it?
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 :)
Please do a video on differentiation under the integral sign
will add it to my list - thanks for the suggestion!
Great ! Thank you !
You're welcome Doktor Klaus!
You are such a great teacher.
Great video sir 👏
Thanks 👍
It's amazing explanation sir!!
Thanks Rekha.
i jst had to subscribe man you solved all my problems of gamma
Awesome - thanks Phumudzo.
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
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 ...
Thank you
You're welcome :)
could someone explain how this could be used to explain the forgotten index of the Fibonacci cube Gamma(n).
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 :)
Nice video
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.
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!
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!
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!
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.
Thanks man.
You're very welcome Tawfique.
Love from India😍😍😍
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 video! It was fun to listen to. But where were the complex numbers?
Great video! I'd just like to point out that the denominator in phi(x) should contain sigma, not sigma squared
The vocabulary you use when describing maths makes it sound very exciting.
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
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
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.
You make esay gamma function for me
thanks
Happy to help Suneel :)
That’s interesting. N-1!
Exponential Integral Gamma function??
Pretty much yes.
Is it arithmetic progression
Thank you , could you cover beta-gamma relationship?
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.
Great explanation. Thank you!
You're very welcome Bekoe.
You said in factorial you multiply all whole numbers less or equal but it must be natural numbers
yes all positive whole numbers
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.
What is this complicated thing used for?
Well, it is easy! All you need is this even more complicated thing that even fewer people understand! Just chuck that in and you are got to go fam
Just watch this impressive Math channel ruclips.net/channel/UCZDkxpcvd-T1uR65Feuj5Yg
oh, that guy.
Whole numbers are 0,1,2,3,4,….. I think what you meant were Natural numbers 1,2,3,4….
RUclips stalls on the crazy eyes at 10:29.
Ha nice spot.
can any one recommend a text book that I can study this
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!
Genius
Thanks Hardik.
love from india
Awesome!
Thanks :)
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
Hi
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