.One can expand this as a slowly converging series in powers of L, but it is very difficult to get enough terms to provide sufficient accuracy at higher latitudes. As we have a very good starting estimate of the inverse, and we are dealing with an analytic function having a simple derivative, the Newton-Raphson root finding algorithm is easy to program and converges quite rapidly.
So consider the implicit equation for t given l, L( t) = l. Define
f(t) = L(t)-l. Then we seek t such that f(t)
= 0. Expand f(t) in a Taylors series with respect to t up to first order.
We wish 
to be that t
for which f(t) = 0 so...
Which we can solve for ...
So filling in for f and f'...
As e is small, we can get an excellent starting estimate from letting e go to zero in the
defining equation (55). Thus 
.