Being generated by a circular loop, the density of radiated momentum
pointing into the direction is independent of the polar
angle . Consequently, that density's spatial integral,
Eq.(66), reduces to

(86) |

The -integration affects only the dotted factors, and they vanish outside a sufficiently large -interval, i.e. . Consequently, integration by parts yields

and

Thus there remain only three non-zero terms in the integral,

The coefficients of the squared terms are

We now take advantage of the fact that the integral, Eq.(87), is independent of the synchronous time . This simplifies the evaluation of the integral considerably because one may assume

without changing the value of the integral. The final outcome is that (i) in each of the expressions, Eqs.(88)-(90), only the last term contributes to the -integral and (ii) the integral assumes a simple mathematical form if one introduces

as the new integration variable. With this scheme one has

The to-be-used integrands have the form

both of which are always less than one in absolute value, even when they get multiplied by . Consequently, one is perfectly justified in saying that

Taking note that only the last terms of Eqs.(88)-(90) give nonzero contribution, apply the limiting form, Eq.(92), to evaluate the integral, Eq.(87). One finds that

(93) |

The value of the integral

implies that the final result is

(94) |