by combining both analytical and numerical computations. In their analysis, the

self-force may be considered to be calculated from

 self lim [Fet() di(p)], (4-25)
 p-p'

where p' is the event on F where the self-force is to be determined and p is an

event in the neighborhood of p', and Fa(p) is related to Q(p) via Eq. (4-1). For

use of this equation, both Fret(p) and F"ir(p) would be expanded into multiple

f-modes, i.e y:, F7 (p) and Z:, F r' (p), respectively, where ire (p) is determined

numerically and Ff (p) determined analytically. In order to determine ~j (p), we

solve Eq. (4-3) using spherical harmonic expansions. The source Q in the equation

is expanded into spherical harmonics, and similarly the field Qret is expanded

 ', r ')(r, t), (4-26)


where (r, t) is found numerically. The individual components '(r, t) in this

expansion are finite at the location of the particle even though their sum, the right

hand side of Eq. (4-26) is singular. Then we have

 qret q V iretv ) (4-27)
 rn

which is also finite. The remaining part ff(p) is determined by a local analysis

of the Green's function in the neighborhood of the particle's world-line. Ref. [9]

provides

 lim j-- (p) (i A B + + + O(f-2), (4-28)
 P P' \ 2 / +
where Aa, Ba and Ca are constants and are generically referred to as R il., rization

Parameters. The remainder is defined as

 D [- lim f (p) + 2)A B --e- (4-29)
 So-P' 20 f -
 =0