Gamma function                                    

Def. Gamma function. The improper integral

This is Euler’s second integral. The integral converges for all positive, real values of x.

Syn. generalized factorial function

Important properties:

For m = 2, formula 4) reduces to 3).

If x is a positive integer n,

5)       Γ(n) = (n - 1)!

Because the gamma function reduces in this special case to (n - 1)! it is often called the generalized factorial function.

Special values

Other definitions of the gamma function.

Euler definition:

valid for all x.

Weierstrauss definition: An infinite product representation

valid for all x, where C is Euler’s constant

Derivative of the gamma function.

Asymptotic expansions for the gamma function

Stirling’s asymptotic series. Either of the two asymptotic expansions

            where B₁, B₂, ...... are the Bernoulli numbers 1/6, 1/30, 1/42, ....... .

If x is a positive integer n and n is large, an approximation is given by Stirling’s formula

Incomplete gamma functions. The incomplete gamma functions are defined by

Important properties:         

References.

1. James/James. Mathematics Dictionary

2. The International Dictionary of Applied Mathematics. D. Van Nostrand Co.

3. Murray R. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)