The Gamma (always written capitalized) function is defined at the definite improper integral:
Gamma(x) = integral, from zero to infinity, of t^(x - 1) exp(- t) dt
Then, it may be shown that the factorial of x is x! = Gamma(x + 1). Hint: reduce the Gamma(x + 1) to the Gamma(x), by employing integration by parts. Then, apply Mathematical Induction.
From 1!. it follows that Gamma(2) = 1. Exercise for the reader: Evalueate Gamma(2) from its defining integral and show directly that Gamma(2) = 1. Prove that 0! = 1. Hint: Evaluate Gamma(1) from its defining integral.
Prove that Gamma(1 / 2) = sqrt( pi ). Hint: Employ the same method used to prove that the Gaussian probability distribution integrates to one:
integral, from minus infinity to infinity, of (1 / sqrt( 2 pi)) exp(x^2 / 2) dx = 1.
Hint: multiply by the same integral along the y-axis. Consider it as an area-integral and evaluate in polar coordinates.
The Gamma function is the easiest non-elementary function. It has been studied extensively. Whole books have been devoted to the Gamma function. We will state two additional relationships:
The Beta (always written capitalized) function of (m, n) is defined as the integral
B(m, n) = integral, from zero to one, of x^(m - 1) (1 - x) ^(n - 1).
It is easy to show that B(m, n) = Gamma(m) Gamma(n) / Gamma(m + n).
It can be shown that
Gamma(n) Gamma(1 - n) = pi / sin(n pi)
Of course, n cannot be an integer. Why? (Rethoric question; to be answered by the reader.).
There are several formulae for evaluating the Gamma function.
Copywrite (c) 1997 R. I. 'Scibor-Marchochi last modified Tuesday 20-th May 1997.