Hey mmzaj.

Have you looked at standard complex analytic techniques to get a taylor series expansion through differentiation?

If the function is entire then all derivatives will exist and you can get a series expansion by using the standard definition of calculating a derivative and evaluate that at the right point.

If there are only a fixed number of singularities for the mapping (in terms of the number of points that produce infinities or non-sensical values) then you can still obtain the series.