Page 384 - Schaum's Outline of Theory and Problems of Advanced Calculus

Gamma and Beta

Functions

THE GAMMA FUNCTION
The gamma function may be regarded as a generalization of n! (n-factorial), where n is any positive
integer to x!, where x is any real number. (With limited exceptions, the discussion that follows will be
restricted to positive real numbers.) Such an extension does not seem reasonable, yet, in certain ways,
the gamma function defined by the improper integral
ð
1 x 1 t
t e dt
ðxÞ¼ ð1Þ
0
meets the challenge. This integral has proved valuable in applications. However, because it cannot be
represented through elementary functions, establishment of its properties take some effort. Some of the
important ones are outlined below.
The gamma function is convergent for x > 0. (See Problem 12.18, Chapter 12.)
The fundamental property

ðx þ 1Þ¼ x ðxÞ ð2Þ
may be obtained by employing the technique of integration by
parts to (1). The process is carried out in Problem 15.1.
Γ(n)
From the form (2) the function ðxÞ can be evaluated for all
5
x > 0 when its values in the interval 1 % x < 2 are known. 4
(Any other interval of unit length will suffice.) The table and 3
graph in Fig. 15-1 illustrates this idea. 2
1
_ 5 _ 4
_ 3 _ 2 _ 1 1 2 3 4 5 n
TABLES OF VALUES AND GRAPH OF THE GAMMA _ 1
FUNCTION _ 2
_ 3
n
_ 4
ðnÞ
1.00 1.0000
_ 5
1.10 0.9514
1.20 0.9182
1.30 0.8975 Fig. 15-1
375
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