The MacLaurin (Taylor series at x=0) series for the arcsin
is a very slowly converging one, especially close to the branchpoints at -1 and 1.
The MacLaurin series for arccos is the same as for arcsin because of
so it has the very same problem.
The series for the arctan on the other side is simpler to compute.
Another good reason: I have it already implemented 😉
The arcsin is related to the atan by
which seems to be of no help at all because the fraction does not get very small and the series for atan is also quite slow near -1 and 1. We can use another relation for this range.
Seems not of much use but we can expand the root
The closer x comes to the branchpoint the larger the fraction, but with
the actual value gets close to zero when x goes close to one and so
which is correct and as intended. The point where is at and at this point .
My cutoff points will be for and for .
No code yet, just twiddling with math, sorry 😉
For the real part of the result is always and the imaginary part . This comes from one of the definitions
I will use this definition for complex arguments.