Inverting series that are flat at zero

The previous post looked at solving the equation

which arises from the Mollweide map projection. Newton's method works well unless is near /2. Using a modified version of Newton's method makes the convergence faster when = /2, which is kinda useless because we know the solution is $theta; = /2 there. When is near /2, the modified Newton's method may diverge. I ended the previous post by saying a series solution would work better when is sufficiently close to /2. This post will flesh that out.

Let x = - 2. Now the task is to solve

for small positive values of y.

The left side is

and so for very small values ofy, 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 iny that gives the solutionx. 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 y^1/3 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.

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