Single-Particle Electrodynamics - Assassination Science

at t = t ret ≡ −R; thus,
R ≡ r − z r ′(t ret ). (6.45)
Using (3.23), we have
z r ′(t) = r ′ + 1 2 t2 ˙v + 1 6 t3¨v − 1 2 t2 (r ′· ˙v) ˙v + 1
24 t4 ...
v − 1 3 t3 (r ′· ˙v)¨v
Thus, by (6.45),
− 1 6 t3 (r ′·¨v) ˙v + 1 2 t2 (r ′· ˙v) 2 ˙v + 1 ....
120 t5 v − 1 8 t4 (r ′· ˙v) v
...
− 1 8 t4 (r ′·¨v)¨v − 1 24 t4 (r ′·... v) ˙v − 1 8 t4 ˙v 2 (r ′· ˙v) ˙v + 1 2 t3 (r ′· ˙v) 2¨v
+ 1 2 t3 (r ′· ˙v)(r ′·¨v) ˙v − 1 2 t2 (r ′· ˙v) 3 ˙v + O(ε 6 ). (6.46)
R = r − r ′ − 1 2 R2 ˙v + 1 6 R3¨v + 1 2 R2 (r ′· ˙v) ˙v − 1 24 R4 ...
v − 1 3 R3 (r ′· ˙v)¨v
− 1 6 R3 (r ′·¨v) ˙v − 1 2 R2 (r ′· ˙v) 2 ˙v + 1 ....
120 R5 v + 1 8 R4 (r ′· ˙v) v
...
+ 1 8 R4 (r ′·¨v)¨v + 1 24 R4 (r ′·... v) ˙v + 1 8 R4 ˙v 2 (r ′· ˙v) ˙v + 1 2 R3 (r ′· ˙v) 2¨v
+ 1 2 R3 (r ′· ˙v)(r ′·¨v) ˙v + 1 2 R2 (r ′· ˙v) 3 ˙v + O(ε 6 ). (6.47)
6.5.2 Length of expressions
Because the expressions from this point become exceedingly long, we shall
omit terms of lower orders in the explicit expressions listed here, where such
an omission leads to a considerable shortening; the full expressions are given
in Section G.6 of Appendix G.
However, the O(ε n ) notation included here still indicates the order retained
in the full expressions, ignoring the fact that terms have been omitted
for typographical sanity.
255