Approximations

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 $a ξ (τ)$ as being close to the straight line, $a a ξ0(τ) = τδ0$ ; 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 $l = 2$ 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 ), $∗ τ → τ = F (τ)$ , to simplify $a ξ (τ )$ and the ‘velocity vector’,

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

G(τ) = 𝔊R (τ ) + i𝔊I (τ),

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