becomes
Qa[e-1] ~ (sin (P) (cos K()2n-p X-n PR, (4-202)
where n = 1, 2 and p 0, 2, hands -1 for a = t,r hands = 0 for a = 0, Only
this form will be taken in the actual calculations of regularization parameters.
To calculate the regularization parameters using Eq. (4-202), first, change
Qak[-1] into the expression in X. From Eq. (4-156) we have
^sin2 (+ 2)(1 (4-203)
sin 2 -J2 o(4-203)
J 2
(r2 2 r2
cos2 o ( J)x r 0 (4-204)
J2
And from Eq. (4-155)
0p1 = (Tr2 + 2) -1/2 -1/2 (S2 + cosSO) -1/2. (4-205)
However, by Eq. (4-171) this can be written as
00
p0 (2 + j2)-1/2x-1/2 (cosO), 6 0. (4-206)
e=o
Then, substituting all the results of Eqs. (4-203), (4-204) and (4-206) into
Eq. (4-202), our Qa[e-1] can be rewritten in a generic form
n oo
Qa[e-l] E E/ (a)X --1/2 E COS 0) (4-207)
n=1,2 m=0 =0
where 3nm(a) depends on n and m as well as on the component index a, with a
dimension ~ 1/R2 for a = t, r and ~ 1/7 for a = 0, 0, and is determined by the
coefficients of the terms in the polynomials PII and PIl of Eq. (4-144). The final
steps of our calculations are taking the integral average of Qa[e-1] over 0 < I < 27
and then taking the coincidence limit -- 0, i.e.
nT oo
Qa [E e-1 o = 1,2() 0 x--1/2) 1. (4-208)
n=-1,2 m=0 =0