The previous post looked at solving the equation
which arises from the Mollweide map projection. Newton’s method works well unless φ is near π/2. Using a modified version of Newton’s method makes the convergence faster when φ = π/2, which is kinda useless because we know the solution is $theta; = π/2 there. When φ is near π/2, the modified Newton’s method may diverge. I ended the previous post by saying a series solution would work better when φ is sufficiently close to π/2. This post will flush that out.
Let x = π − 2θ. Now the task is to solve
for small positive values of y.
The left side is
and so for very small values of y, and thus very small values of x, we have
If this solution is not sufficiently accurate, we can invert the power series above to get a power series in y that gives the solution x. However, the Lagrange inversion theorem does not apply because the series has a zero derivative at 0. Instead, we have to use Puiseux series inversion, looking for a series in y1/3 rather than a series in y. From the Puiseux series we can see that
is a more accurate solution. For even more accuracy, you can compute more terms of the Puiseux series.