### 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,
;Update 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, Equation (79), , to simplify 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 , with the decomposition

the ‘velocity’ transforms as

One can easily check that by the appropriate choice of we can make and
orthogonal, i.e.,

and by the choice of , the can be normalized to one,
The remaining freedom in the choice of is simply an additive complex constant, which is used
shortly for further simplification.

We now write , which, with the slow motion approximation, yields, from the
normalization,

Update From
the orthogonality, we have
i.e., is second order. Since is second order, is a constant plus a second-order
term. Using the remaining complex constant freedom in , the constant can be set to zero:

Finally, from the reality condition on the , Equations (94), (97) and (96) yield, with
and treated as small,

Within this slow motion approximation scheme, we have from Equations (109) and (110),
or, to first order, which is all that is needed,

We then have, to linear order,