@@physicsandmathlectures3289 Thanks for sharing this but it didn't really explain where e ^x comes from with respect to a factorial of a non-integer so I don't think anyone can get a full understanding just from this right??.can you elaborate on that on that we know exactly why the Gama function is defined in this way? Don't know if you already covered that sorry..
I didn't understand part of your explanation where you explained gamma(n+1) on the second line of integration 6:51 someone please help me as soon as possible
Great video. It would be nice to add a proof that gamma(0)=1 which, given your other proof, would establish that gamma(n+1)=n! for the positive integers.
@@koush69 Looking it up on Wikipedia, you are correct that Gamma(0) can't be defined to be 1 without breaking the continuity of the Gamma function so it doesn't match the factorial function for all natural numbers. One would need to add a proof that Gamma(1)=1 to establish Gamma(n+1)=n! for the positive integers. It's also interesting to see that Gamma is defined for all complex numbers except the non-positive integers. I guess you mean Gamma(n) is undefined when n
@@physicsandmathlectures3289 wait I don't think this totally.clear..why isnt z the dummy variables since I'm I'm factorial you multiply n by itself n times so it shouldbe z^z-1 Terence is no need for another variable x ..thst needlessly complicates things..see what zi mean..
The Gamma function is the only function that has these 3 properties:For x>0: f(x+1) = x f(x) and f is log convex ( ln(f) is convex) and f(1)=1 as proven by Harald Bohr (Niels Bohr's brother) and Johannes Mollerup.
By your logic at 7:26, shouldn't Gamma(z+1) be equal to zGamma(z) - UV after using integration by parts? Does the UV piece go away because evaluated at those bounds, you end up dividing by infinity?
Kinda late but if u think about the answer can be seen in the example at the start with 1, 2 , 3, 4 when they get plotted on the graph. If you just think about it following the rules of factorials with integers , each factorial must contain the multiples used in the factorial below it. For example 3! contains the factorial 2! Because 3! Is written as (3 • 2 • 1) and we know 2! is ( 2 • 1). Thus since we can see that 2! Is contained within 3! We can then call 3 “n” and see that the factorial function for integers can be expressed as “n • (n-1)!” since we just covered that 3! Is (3)(2)(1) or 3 • (3-1)!
This video is simply an introduction he didnt just make up this formula. Getting the formula takes a bit because it requires integral calculus. There are videos showing how to find the formula through integration if youre interested.
I m working on it; i consider factorial as special case when f(x)=x; functional factorial is: f!(x+1)=f(x).f!(x) ; not yet finished; many thing appeared to on way.
All of these gamma function videos are RIGHT, but damn near all of them miss the entire point. None of y'all even address let alone answer the question HOW WOULD ANYONE EVER COME UP WITH THAT. As long as the definition is random-handed-down-by-god, no actual learning has happened.
That's how most math is taught especially something like calculus. When you start learning calc you just made to memorize limits and formulas and standard derivatives without any context. No one takes the effort to show an intuitive proof for math. I'm a highschool student and all the calc that I have learnt is almost entirely self taught only being supplemented by few TH-cam vids Wolfram Alpha textbooks. None of my teachers take the time to explain in an intuitive sense what fuck a derivative or an integral is and how people came up with it. It's a big pain the ass and especially in my country college entrance tests are extremely competitive so you have to cram in as many formulas as possible without learning anything beyond the elementary concept it represents. It's truly a tragedy.
Очень интересно... но где то подвох... Что мы знаем о факториалах... Для начала мы знаем что факториал следующего числа равен факториалу предыдущего числа умноженному на это самое следующее число... N!= (N-1)!×N или по другому... факториал предыдущего числа равен факториалу следующего числа деленному на это самое следующее число... N!=(N+1)!/(N+1) есть еще вид (N+1)!= N!×(N+1)... значит (N-1)!=N!/N и N=N!/(N-1)! При N=1 получаем 0!=1!/1 и 1=1!/0! При N=0 получаем (-1)!=0!/0 и 0=0!/(-1)! При N=(-1) получаем (-2)!=(-1)!/(-1) и (-1)=(-1)!/(-2)! При N=(-2) получаем (-3)!=(-2)!/(-2) и (-2)=(-2)!/(-3)! При N=(-3) получаем (-4)!=(-3)!/(-3) и (-3)=(-3)!/(-4)! При N=(-4) получаем (-5)!=(-4)!/(-4) и (-4)=(-4)!/(-5)! Видим что вычисление положительных факториалов по действию очень похоже на действие возведения в степень... только множители различные... Исходя из полученных формул отрицательный факториал берется не только от отрицательного значения но и имеет смысл обратных значений для положительных факториалов N... Во всяком случае вполне возможно N!=(N+1)!/(N+1) 0!=1!/1=1 (-1)!=0!/(0)=1/(0)= 1 неделённая единица (-2)!=(-1)!/(-1)= 1/(-1)= -1 (-3)!=(-2)!/(-2)=(-1)/(-2)= 1/2 (-4)!=(-3)!/(-3)=(1/2)/(-3)= -1/6 (-5)!=(-4)!/(-4)=(-1/6)/(-4)= 1/24 (-6)!=(-5)!/(-5)=(1/24)/(-5)= -1/120... Интересно что получаются обратные значения Гамма функциям от положительных значений когда Г(N+1)=N! Г(N+1)=N×Г(N)=N×(N-1)! Немного неожиданно... Получается что для отрицательных Г(-(N+1))=1/Г(N+1)=1/N! Но есть "проблема" со знаком... Видим что постоянно через один изменяется знак при делении "факториалов" от отрицательных значений... Предположу что нужно брать для отрицательных значений N значение по модулю (а для обобщения и для положительных значений N...) N!=(N+1)!/|N+1| (N-1)!=N!/|N| 0!=1/1=1 (-1)!=0!/0=1/0= 0 (относительный ноль) или безотносительно единица неделённая что более верно... Тогда следует (-2)!= (-1)!/|-1|=1 (-3)!=(-2)!/|-2|=1/2 (-4)!=(-3)!/|-3|=1/6 (-5)!=(-4)!/|-4|=1/24... Как видим получаем обратные величины факториалов для положительных значений N... но еще идет сдвиг на один ход относительно факториалов для положительных значений N... Смею предположить что отрицательные факториалы должны считаться по формуле N!=(N+1)!/|N|... Тогда (-1)!=0!/|-1|=1/1=1 (-2)!=(-1)!/|-2|=1/2 (-3)!=(-2)!/|-3|=1/6 (-4)!=(-3)!/|-4|=1/24 (-5)!=(-4)!/|-5|=1/120... и получается что эти значения численно равны коэффициентам для нахождения "обратного факториала"... Кстати по этой же формуле получается 0!=1!/0=1/0=1 единица неделённая что наверное будет более верно... Если уж быть совсем дерзким и исходить из того что график этих значений должен бы быть хоть немного математически красив то возможно факториалы от отрицательных значений должны бы быть и сами отрицательными... Но я пока не нахожу физического смысла отрицательным значениям факториалов... (самим факториалам от отрицательных чисел смысл проявился очень явно)... к тому же придется признать что тогда при этом 0!=1/0=0 равен относительному нулю... Но это пока мои личные фантазии... и в этом надо сначала разобраться... а перед этим хорошенько подумать... Мне все же ближе "вариант с модулями"...
You have to do something about your microphone. It's capturing the sounds of every minute saliva slosh and tongue flick inside your mouth. I had to mute it.
You sir, are a great teacher. I understood that without even pausing. That's rare. Thank you so much
dude i love his videos, but the guy didn't even do anything. He just motivated it and gave the definition of the gamma function
@@metuphys5611 there is nothing wrong I said.
@@actualBIAS yeah, there was nothing wrong, i totally agree. just your selection of video to comment on was kind of off. That's all.
@@metuphys5611 If you think so...
Dude he did not teach anything. He just restated facts.
Wonderful vid but I had a chuckle when you included (10/2)! as something that now has a meaning
Started learning this this semester, thank you sir
No problem!
Kelas berapa kak?
@@physicsandmathlectures3289 Thanks for sharing this but it didn't really explain where e ^x comes from with respect to a factorial of a non-integer so I don't think anyone can get a full understanding just from this right??.can you elaborate on that on that we know exactly why the Gama function is defined in this way? Don't know if you already covered that sorry..
I entered the weeds when the green pen came out.
I didn't understand part of your explanation where you explained gamma(n+1) on the second line of integration 6:51 someone please help me as soon as possible
WHAT AN EXPLANATION, THANK YOU MAN :)
the 10/2 factorial is indeed very weird lol
Great video. It would be nice to add a proof that gamma(0)=1 which, given your other proof, would establish that gamma(n+1)=n! for the positive integers.
Gamma(0) is undefined we can't find Gamma(n), where n
@@koush69 Looking it up on Wikipedia, you are correct that Gamma(0) can't be defined to be 1 without breaking the continuity of the Gamma function so it doesn't match the factorial function for all natural numbers. One would need to add a proof that Gamma(1)=1 to establish Gamma(n+1)=n! for the positive integers. It's also interesting to see that Gamma is defined for all complex numbers except the non-positive integers. I guess you mean Gamma(n) is undefined when n
Great introduction.
Outstanding job, your lecture couldn't have been any clearer!
Please make a video on Riemann's hypothesis.....
Awesome video and channel, very helpful, thank you!
Glad it was helpful!
@@physicsandmathlectures3289 wait I don't think this totally.clear..why isnt z the dummy variables since I'm I'm factorial you multiply n by itself n times so it shouldbe z^z-1 Terence is no need for another variable x ..thst needlessly complicates things..see what zi mean..
The Gamma function is the only function that has these 3 properties:For x>0: f(x+1) = x f(x) and f is log convex ( ln(f) is convex) and f(1)=1 as proven by Harald Bohr (Niels Bohr's brother) and Johannes Mollerup.
Any general gamma function so we found out function out of function 🤣🤣🤣🤣🤣
Out here trying to understand Coast Contra’s bar about square root of pi
Interesting! 😊
A quick question: why do we shift by 1 in the definition of the Gamma function?
Great video!
By your logic at 7:26, shouldn't Gamma(z+1) be equal to zGamma(z) - UV after using integration by parts? Does the UV piece go away because evaluated at those bounds, you end up dividing by infinity?
why not have z instead of z-1? wouldn't it then be r(z)=z! instead of r(z)=(z-1)!?
Very good explanation.Thank you
Excellent and clear explanation
Glad it was helpful!
Just needed the Gama(n+1)=n!
Thx
What about n!!, n!!!, ... (double, tripple,... factorial?
Why isn’t the gamma function just x^z instead of x^z-1?? Then Gamma(n) = n!
On thiss🗣️🎶 minuss🗣️🎶 x's🗣️🎶
if I pass tomorrow in my test I will subscribe to this channel 😭wish me luck
Did you pass
can someone say how he differentiate and integrate in the integral at same time ?
Integration by parts method
Helpful.
Glad to hear!
Why can n! also be n(n-1)! ? Keep in mind I’ve never covered factorials in any of my math classes what a shame
Kinda late but if u think about the answer can be seen in the example at the start with 1, 2 , 3, 4 when they get plotted on the graph.
If you just think about it following the rules of factorials with integers , each factorial must contain the multiples used in the factorial below it. For example 3! contains the factorial 2! Because 3! Is written as (3 • 2 • 1) and we know 2! is ( 2 • 1). Thus since we can see that 2! Is contained within 3! We can then call 3 “n” and see that the factorial function for integers can be expressed as “n • (n-1)!” since we just covered that 3! Is (3)(2)(1) or 3 • (3-1)!
So you just pulled the "formula" out of your hat.
This video is simply an introduction he didnt just make up this formula. Getting the formula takes a bit because it requires integral calculus. There are videos showing how to find the formula through integration if youre interested.
I m working on it; i consider factorial as special case when f(x)=x; functional factorial is: f!(x+1)=f(x).f!(x) ; not yet finished; many thing appeared to on way.
You missed explanations … sadly I didn’t get what you are trying to say
Sorry to hear that! I'd love to hear any suggestions or specific criticisms you have.
All of these gamma function videos are RIGHT, but damn near all of them miss the entire point.
None of y'all even address let alone answer the question HOW WOULD ANYONE EVER COME UP WITH THAT.
As long as the definition is random-handed-down-by-god, no actual learning has happened.
That's how most math is taught especially something like calculus. When you start learning calc you just made to memorize limits and formulas and standard derivatives without any context. No one takes the effort to show an intuitive proof for math. I'm a highschool student and all the calc that I have learnt is almost entirely self taught only being supplemented by few TH-cam vids Wolfram Alpha textbooks. None of my teachers take the time to explain in an intuitive sense what fuck a derivative or an integral is and how people came up with it. It's a big pain the ass and especially in my country college entrance tests are extremely competitive so you have to cram in as many formulas as possible without learning anything beyond the elementary concept it represents. It's truly a tragedy.
Hi sir! We can't see papa
Can't see anything 🥲
Unless God gave you the definition and said "Here it is!" without any proof of how it came to be, then you explained nothing.
What the hell is this comment supposed to mean?
Tf you trying to say
@@Harwey-lz4gp THATS EXACTLY WHAT I SAID
There is no discussion, we made this up and we decided it is like this.
It was a nice intro though
Why real part (z)>0
Ten halves factorial...
I needed a solution for upper and lower incomplete gamma function to derive poisson and gamma distribution, this doesn't help.
Очень интересно... но где то подвох...
Что мы знаем о факториалах...
Для начала мы знаем что
факториал следующего числа равен факториалу предыдущего числа умноженному на это самое следующее число...
N!= (N-1)!×N
или по другому... факториал предыдущего числа равен факториалу следующего числа деленному на это самое следующее число...
N!=(N+1)!/(N+1)
есть еще вид (N+1)!= N!×(N+1)...
значит (N-1)!=N!/N и N=N!/(N-1)!
При N=1 получаем 0!=1!/1 и 1=1!/0!
При N=0 получаем (-1)!=0!/0 и 0=0!/(-1)!
При N=(-1) получаем (-2)!=(-1)!/(-1) и (-1)=(-1)!/(-2)!
При N=(-2) получаем (-3)!=(-2)!/(-2) и (-2)=(-2)!/(-3)!
При N=(-3) получаем (-4)!=(-3)!/(-3) и (-3)=(-3)!/(-4)!
При N=(-4) получаем (-5)!=(-4)!/(-4) и (-4)=(-4)!/(-5)!
Видим что вычисление положительных факториалов по действию очень похоже на действие возведения в степень...
только множители различные...
Исходя из полученных формул отрицательный факториал берется не только от отрицательного значения но и имеет смысл обратных значений для положительных факториалов N...
Во всяком случае вполне возможно
N!=(N+1)!/(N+1)
0!=1!/1=1
(-1)!=0!/(0)=1/(0)= 1 неделённая единица
(-2)!=(-1)!/(-1)= 1/(-1)= -1
(-3)!=(-2)!/(-2)=(-1)/(-2)= 1/2
(-4)!=(-3)!/(-3)=(1/2)/(-3)= -1/6
(-5)!=(-4)!/(-4)=(-1/6)/(-4)= 1/24
(-6)!=(-5)!/(-5)=(1/24)/(-5)= -1/120...
Интересно что получаются обратные значения Гамма функциям от положительных значений когда
Г(N+1)=N!
Г(N+1)=N×Г(N)=N×(N-1)!
Немного неожиданно...
Получается что для отрицательных Г(-(N+1))=1/Г(N+1)=1/N!
Но есть "проблема" со знаком...
Видим что постоянно через один изменяется знак при делении "факториалов" от отрицательных значений...
Предположу что нужно брать для отрицательных значений N значение по модулю (а для обобщения и для положительных значений N...)
N!=(N+1)!/|N+1| (N-1)!=N!/|N|
0!=1/1=1
(-1)!=0!/0=1/0= 0 (относительный ноль)
или безотносительно единица неделённая что более верно...
Тогда следует (-2)!= (-1)!/|-1|=1
(-3)!=(-2)!/|-2|=1/2
(-4)!=(-3)!/|-3|=1/6
(-5)!=(-4)!/|-4|=1/24...
Как видим получаем обратные величины факториалов для положительных значений N...
но еще идет сдвиг на один ход относительно факториалов для положительных значений N...
Смею предположить что отрицательные факториалы должны считаться по формуле
N!=(N+1)!/|N|...
Тогда
(-1)!=0!/|-1|=1/1=1
(-2)!=(-1)!/|-2|=1/2
(-3)!=(-2)!/|-3|=1/6
(-4)!=(-3)!/|-4|=1/24
(-5)!=(-4)!/|-5|=1/120...
и получается что эти значения численно равны коэффициентам для нахождения "обратного факториала"...
Кстати по этой же формуле получается
0!=1!/0=1/0=1 единица неделённая
что наверное будет более верно...
Если уж быть совсем дерзким и исходить из того что график этих значений должен бы быть хоть немного математически красив то возможно факториалы от отрицательных значений должны бы быть и сами отрицательными...
Но я пока не нахожу физического смысла отрицательным значениям факториалов...
(самим факториалам от отрицательных чисел смысл проявился очень явно)...
к тому же придется признать что тогда при этом 0!=1/0=0 равен относительному нулю...
Но это пока мои личные фантазии...
и в этом надо сначала разобраться...
а перед этим хорошенько подумать...
Мне все же ближе "вариант с модулями"...
(10/2)! ???
5!, 120
You have to do something about your microphone. It's capturing the sounds of every minute saliva slosh and tongue flick inside your mouth. I had to mute it.