APPENDIX A
HYPERGEOMETRIC FUNCTIONS AND REPRESENTATIONS OF
REGULARIZATION PARAMETERS
In Section 4.4 we define
X 1 a sin2 4
(Al)
(A2)
S+ J2'
And we use
a sin2 )-P d~
-) a sin2 -P
In particular, for the cases p = and p -
F1/2 1 ,
(2 21-
and
F-1/2 -21 (
2 1 ( ; 1,) a
2 (
-K(a)
i7
(A3)
Swe have the following representations
(A4)
(A5)
where K(a) and E(a) are called complete elliptic integrals of the first and second
kinds, respectively.
If we take the derivative of F1/2 with respect to k
V via Eq. (A3), we
obtain
F1/2 F3/2
k k
(A6)
with
2 j/2
2/ /
71-J
F1/2
Ok
1 1
2 21; a)
2' 2