Gamma Function

The Gamma (always written capitalized) function is defined as the definite improper integral:

Gamma(x) = integral, from zero to infinity, of t^(x - 1) exp(- t) dt

The factorial of a non-negative integer n is defined recursively as

n! = n * (n - 1)! if n > 0
= 1 otherwise.

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: Evaluate 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.

The Stirling asymptotic formula for the Gamma function is

ln(Gamma(x)) = x * (ln(x) - 1) + 0.5 * ln(2 * pi / x) + ln(1 + 1 / (12 + x) + 1 / (288 * x * x) - 139 / (51840 * x * x * x) - 571 / (2488370 * x * x * x * x) + ...)

This formula provides a good approximation when x > 30. The formula is quoted in numerous places. The only derivation known to me is in Advanced Calculus by Franklin. This comment had been written Monday 14-th August 2000. The WWW (= World Wide Web) grows as fast as the gamma function! Now, the Google search-engine provides a surfeit of hits on "Stirling's formula". Here are two links to derivations of the Stirling's formula: first second.