What is the Gamma Function? What is Gamma of 4?

Date: 05/28/98 at 23:35:29
From: dan masdeo
Subject: function gamma

What is the function gamma? Specifically, what is gamma of 4?

Date: 05/29/98 at 05:22:34
From: Doctor Anthony
Subject: Re: function gamma

The Gamma function is defined as:

   G(n) = INT(0 to infinity)[x^(n - 1) e^(-x) * dx]

Now, integrating by parts:

   G(n) = [x^(n - 1) * (-e^(-x))] 
          + INT[(n - 1)x^(n - 2) * e^(-x) * dx]

The first bracket is zero at both ends when limits infinity and 0 are 
put in.

So:

   G(n) = (n - 1) INT[x^(n - 2) * e^(-x) * dx]

   G(n) = (n - 1) * G(n - 1)

When n is a positive integer, this defines the factorial function:

   G(n) = (n - 1) * (n - 2) * (n - 3) ... 3 * 2 * 1 * G(1)

and:

   G(1) = INT[e^(-x)] from 0 to infinity 

        = -[e^(-x)]  from 0 to infinity

        = -[0 - 1]

        = 1

It follows that G(n) = (n - 1)!

It follows also that G(1) = (1 - 1)! = 0!

We have seen that G(1) = 1, so then 0! = 1 

To find Gamma of 4, we see that G(4) = 3! = 6

-Doctor Anthony, The Math Forum
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