### 3.3 Approximations

Due to the difficulties involved in the intrinsic nonlinearities and the virtual impossibility of exactly
inverting arbitrary analytic functions, it often becomes necessary to resort to approximations. The basic
approximation will be to consider the complex world line as being close to the straight line,
; deviations from this will be considered as first order. We retain terms up to second order,
i.e., quadratic terms. Another frequently used approximation is to terminate spherical harmonic expansions
after the terms.
It is worthwhile to discuss some of the issues related to these approximations. One important issue is
how to use the gauge freedom, Eq. (3.8), , to simplify and the ‘velocity vector’,

A Notational issue: Given a complex analytic function (or vector) of the complex variable , say
, then can be decomposed uniquely into two parts,

where all the coefficients in the Taylor series for and are real. With but a slight extension
of conventional notation we refer to them as real analytic functions.

With this notation, we also write

By using the reparametrization of the world line, via , we choose , so that
(dropping the ) we have
Finally, from the reality condition on the , Eqs. (3.23), (3.26) and (3.25) yield, with
and treated as small,

Within this slow motion approximation scheme, we have from Eqs. (3.40) and (3.41),
or, to first order, which is all that is needed,

We then have, to linear order,