Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.16 GAMMA FUNCTIONS

Definition: Gamma Function The gamma function is the following definite integral:

(B.84)

where (though we only consider real values in this book).

It has the following property: , and for :

(B.85)

For noninteger numbers a, the gamma function can be viewed as an extension of the factorial function. A plot of is provided in Figure B.11. Observe that for negative integers, the function alternates between , and for positive integers it is the factorial function (with argument shifted down by one), as given in (B.85). It can also be shown that for :

Figure B.11 Gamma function .

(B.86)

with and . The Euler–Masheroni constant (described later) is related to the gamma ...