DERIVING THE GAMMA FUNCTION – Combining Feynman integration and Laplace Transforms
DERIVING THE GAMMA FUNCTION – Combining Feynman integration and Laplace Transforms

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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