1.8 The Incomplete Beta Function
17
and limited to positive real values for “a, b” and 0 < x ≤ 1. The incomplete Beta
function is related to the hypergeometric function 2 F 1 [3] via
B z (a, b) =
z a
a
2 F 1 (a, 1 − b; a + 1; z)
and
(1.44a)
=
z a (1 − z) b
a
2 F 1 (a + b, 1; a + 1; z).
(1.44b)
The computation will be based on these relations to the hypergeometric function
and its transformation properties (8.3).
1.8.2 Computational Aspects
The computation of the incomplete Beta function is based on a series expansion of
the hypergeometric function, see Chap. 8. Because this expansion only converges
for |z| < 1 argument transformations are necessary to cover the complete complex
space.
The hypergeometric function is defined as
2 F 1 ( ˜
a, ˜
b; ˜
c; z) =
∞
n=0
( ˜
a) n ( ˜
b) n
( ˜
c) n
z n
n!
|z| < 1
(1.45)
=
∞
n=0
t n z
n
(1.46)
with (· · · ) n the Pochhammer symbol, Eq. (1.8). t n can be iteratively computed
t 0 = 1, t n+1 =
( ˜
a + n)( ˜
b + n)
( ˜
c + n)(n + 1)
t n .
(1.47)
Case 1 Due to Eq. (1.44a) we set
˜
a = a, ˜
b = 1 − b, and ˜
c = a + 1,
(1.48)
and thus we get from Eq. (1.47)
t 0 = 1, t n+1 =
(a + n)(1 − b + n)
(a + 1 + n)(n + 1)
t n .
(1.49)
Therefore the series expansion becomes finite and hence exact for 1 − b = −n ⇒
b = n + 1, and if b > −a infinite for a = −n − 1 with n positive integer and z
arbitrary. The computation of B z (a, b) converges only within the unit circle |z| < 1
for arbitrary a, b.