Undergrad Can anyone explain the Gamma function to me?

[Frank Li]

Γ(n) = ∫x→∞ t^n-1 e^-t dt?

 

[jedishrfu]

Can you be more explicit? Like where did you find it? What it's used for?

It looks like the gamma function.

https://en.m.wikipedia.org/wiki/Gamma_function

 

[Frank Li]

Can you be more explicit? Like where did you find it? What it's used for?

It looks like the gamma function.

https://en.m.wikipedia.org/wiki/Gamma_function

I was watching about factorials on Youtube channel by the Numberphile, a topic named "0! = 1". Inside that video, they mentioned about this function, and I would like to look deeper into this topic.

 

[jedishrfu]

Yes, I remember that video.

 

[Frank Li]

Yes, I remember that video.

Yeah, the end of that one.

 

[jedishrfu]

There are some other videos on YouTube that get into more detail about the function and it's uses



Basically though, it came about as mathematicians try to extend factorials to work with rational numbers and beyond which is a common theme in math. Find a pattern and keep extending it outward i.e. Generalizing it more and then prove that it works in the new contexts.

Creativity in action.

 

[MAGNIBORO]

You must learn calculus to understand this function because The gamma function is a function based on an integral, But in other words it is only the area under the curve of a "set of function"
If with a program you could visualize the function $x\, e^{-x}$ And measure the area under the curve from 0 to infinity,
You would find that the area would equal 1!=1 (like a 1x1 square)
If you did the same with the function $x^{2}\, e^{-x}$ the area is 2!=2
$x^{3}\, e^{-x}$ the area is 3!=6
And so on.
as we can measure the area under the function curve as $x^{1/2}\, e^{-x}$
we can say that $\left ( \frac{1}{2} \right )!= \frac{\sqrt{\pi} }{2}$

 

[Frank Li]

There are some other videos on YouTube that get into more detail about the function and it's uses



Basically though, it came about as mathematicians try to extend factorials to work with rational numbers and beyond which is a common theme in math. Find a pattern and keep extending it outward i.e. Generalizing it more and then prove that it works in the new contexts.

Creativity in action.

Thanks that video is so helpful!