If the eccentricity e is small, Kepler’s equation for the eccentric anomaly ψ as a functio...

If the eccentricity e is small, Kepler’s equation for the eccentric anomaly ψ as a function of ωt, Eq. (3.76), is easily solved on a computer by an iterative technique that treats the e sin ψ term as of lower order than ψ. Denoting ψn by the nth iterative solution, the obvious iteration relation is

Using this iteration procedure, find the analytic form for an expansion of ψ in powers of e at least through terms in e3.