How can I express this complex function as a series?
Log[ (1 - E^((I Pi (1 - a))/(b - a)) z)/ (1 - E^(-((I Pi (1 - a))/(b - a))) z) ]
z is the complex number,
b are real numbers,
a < 1 and
b > 1.
Let us define
Then, in order to get the Series expansion coefficient to any order
(* -((2 I sin((π (a-1) n)/(a-b)))/n) *)
If you want to check
(* O[z^16] *) for instance.
Note that you could do the Taylor expansion by hand
and identify the prefactor of