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 Check out our web site! http://mathforum.org/dr.math/ |
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