Website owner: James Miller
Gamma function
Def. Gamma function. For positive values of x, the Gamma function is defined by the improper integral
This is Euler’s second integral. It 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 convenient to view it as a generalized factorial function.
Special values
Definition of the Gamma function for negative values of the argument. The gamma function can be defined for negative values of the argument x by using the formula
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)
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