1550251515-Classical_Complex_Analysis__Gonzalez_

630 Chapter 8

poles, we may use (8.20-33) to generalize the definition of the B-function

to the Cartesian product C x C except at the points (z, ()where one of the

coordinates, or both, are zero or a negative integer. Thus in what follows

we shall write

B( 1") = I'(z)I'(()

z,., I'(z+()' z,( =I-o,-1,-2, ... (8.20-34)

Poin,ts where z + ( = -n (n = 0, 1, 2, ... ) but z, ( =f:. O, -1, -2, ... are
zeros of I/I'(z + (), so they are also zeros of B(z, (). With this extension,
properties 2 through 6 are valid in general, except at those points where
the functions involved are undefined.
Note The method of proof of formula (8.20-32) for positive real values of
the variables is due to T. S. Nanjundiah (27].
Formula (8.20-34), which shows the connection between the beta and the
gamma functions, makes possible the evaluation of a number of important
definite integrals in terms of values of the r-function.

Examples

1. To evaluate I = fo
   1
   dx I vl -x^3 • By letting x = t^113 ' dx =^1 /ar^213 dt
   we find that

I= 1/a 11 r2/a(l - t)-1/2dt = 1/aB(1/a, %)

= 1 / r(1fa)r(1/^2 ) ,..., 1 4043

3 r(s/6) ,...,.

2. To evaluate I = f 0 -rr /^2 ~d(}. Here we have

I= {-rr/

2

sin° Ocos^112 OdO =^1 / B(^1 /^3 1) = ~ I'(%)I'(

3

/4)

lo 2 2• /4 2 r(s/4)

~ 1.1981
Further properties of the B-function are given in Exercises 8.9, problems
7 and 8.

8.21 The Factorial Function

Definition 8. 7 For any complex number a we define

(a)n = a(a + l)(a +

1

2) ···(a+ n -1)

{

(a)o = 1

(a)-n = -:-----:-)(-,-----,-----..,...

(a-1 a-2) .. ·(a-n)

for n = 1, 2, 3, ...

for a =f:. 1, 2, ... , n

(8.21-1)