i,. ,,lar, remainder, which is the difference between the retarded field and singular

field, is believed to be smooth and well behaved on the world-line of the particle,

and is summed over all modes to provide the self-force or the radiation reaction

effect. The mode-decomposed solution of the retarded field is obtained numerically

while the mode-decomposed piece of the singular field is determined in a purely

analytic manner. The latter is generically referred to as BR.l, ,lri, .: -i.:..n Parameters.

 In this ('!h plter we deal with the self-force of a scalar charge orbiting a

Schwarzschild black hole. Section 4.1 introduces a recent method to split the

retarded scalar field in curved spacetime -i '-.- -1. 1 by Detweiler and Whiting

[16]. This follows Dirac's idea in his flat spacetime problem [2], and gives good

interpretations not only in the singular behavior of the retarded field, but also in

its differentiability. In Section 4.2 we give the overview of the mode-sum method

originated by Barack and Ori [9], and present the analytic results for the singular

field, i.e. the regularization parameters, which were obtained by Kim [17]. The

regularization parameters are locally well defined and should describe the singular

behavior and the differentiability of the field precisely. Higher order expansions of

the singular field will generate higher order regularization parameters, and their use

in the mode sums for the self-force calculation will result in more rapid convergence

and more accurate final results. To facilitate the computations of the regularization

parameters, an in-depth analysis of the local spacetime would be demanded, and in

Section 4.3 we develop an elaborate perturbation analysis of the local geometry for

this purpose. Then, Section 4.4 is devoted to the calculations of the regularization

parameters. These results are then combined with the numerical computations of

the retarded field to provide the self-force ultimately. This final task is done in the

Section 4.5.