According to Appendix D of Ref. [18], we have the following Legendre polyno-
mial expansions of (62 + 1 cos )--1/2: for p > 1
(62 + CS ) -"-1 2p-12p 1) [1 + O(M)] P(cos ), 6 0, (4-170)
t o
and for p = 0
(62 COSO) -12 = [2+ O(M)] P(cos), ---0. (4-171)
t 0
Then, by Eq. (4-170) for p = 1, and by Eqs. (4-171) and (4-157), in the limit 6 0
(equivalently A 0) Eq. (4-169) becomes
lim Qt[ -2 sgn(A) q2r C P COS
Ao 2 2 + J )/2=0 2Y
q2EJX-3/2 cos 4 D P(cos ). (4-172)
(r2 + J2)3/2 a o
Then, we integrate limao Qt[c-2] over 4 and divide it by 27r (henceforth, we
denote this process by the angle brackets "()")
olim Qt[-2]) sgn(A) q2 yro (X-1) P(COS ), (4-173)
(r2 + 2 2 3J 0 2
where we exploit the fact that (-3/2 cos )} = 0 to get rid of the second term in
Eq. (4-172).2 Appendix C of Ref. [18] provides (X-1) 211 (1, 1; 1; a) F
(1 a)-1/2, where a J2 (r2 + J2). Substituting this into Eq. (4-173), the
regularization parameter At is the coefficient of the sum on the right hand side in
a
2 Or alternatively, one can use the argument a- P(cos 6) = 0 as 8 -+ 0, to
show that this part does not survive at the end.