I am wondering if there is an efficient method to compute the inverse of a complex function numerically. More precisely I have a certain amount of points $x_1,\ldots,x_n$ in the upper complex half plane and I want to evaluate a certain conformal function on the points. However, I don´t know this function explicitly, but its inverse: $$f(z)=(z+c_1)^{1-\alpha}(z-c_2)^{\alpha},$$ where $c_1,c_2,\alpha \in \mathbb{R}$, and $c_1,c_2 > 0, 0<\alpha<1$ (this conformal map sends the upper half plane to the upper half plane minus a line segment from $0$ to a point in the half plane). I guess I can solve the problem by simply using Newtons method for all the points (evaluating the zero of $f(z)-x_k$). But since the number of points can get very large, I am wondering if there is a more efficient way of doing this. I thought of some kind of complex inverse interpolation, but I have no idea how to do this precisely and if one can do it in an efficient way.
Any comment on this is highly appreciated.