or using Eq. (A4)
aK K 7 F3/2
+ (A7)
ak k 2 k
However, Ref. [36] shows that
K E K
(A8)
ak k(1- k2) k
Thus, by comparing Eq. (A7) and Eq. (A8) we find the representation
2 E 2E
F3/2 (A9)
7 1 k2 7 1 a
Further, we can also find the representation for F5/2. First, taking the deriva-
tive of F3/2 with respect to k Va via Eq. (A3) gives
OF3/2 3F3/2 3F5/2
ak k k
Also, using Eq. (A9) together with Eqs. (A3)-(A5), another expression for the same
derivative is obtained solely in terms of complete elliptic integrals
OF3/2 2 (1 + k)E (1 k2)
(All)
kk l k(1 k 2)2
Then, by Eqs. (A9), (A10) and (All) we find
2 2 (2 a) E K
F5/ 2 2 (2- ) (A12)
37r (1 a)2 1 a
Now, using Eqs. (A4), (A9) and (A12), we may rewrite the non-zero Ba
regularization parameters, Eqs. (4-37)-(4-39) in Section 4.2 as
2 Er [K(a) 2E(a)]
Bt = (A13)
S(1 + J2/r2)3/2 (A
q2 (2 2E2) K(a) + (j2 + E2) E(a)
,, (1 2M/ro) (1 + J2/r2/2