I have the following function that maps an ellipse in $z$-plane into a unit circle in $\zeta$-plane:
$$z = \sum_{n=0}^{N} \alpha_n\zeta^{n+1}$$
where $\alpha_n = a_n + ib_n$ is a constant, and $\zeta = e^{i\theta} = \cos\theta + i\sin\theta$ where [$\theta = 0...2\pi$].
I managed to obtain the constant $\alpha_n$ through iterative processes to transform an ellipse into a unit circle. The result is shown as the blue curve in the plot.
where the following constants $\alpha_n$ has been used to transform the ellipse into a unit circle (blue curve in the figure),
1.5000 + 0.0000i
-0.0000 - 0.0000i
-0.5000 + 0.0000i
-0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 - 0.0000i
-0.0000 + 0.0000i
-0.0000 + 0.0000i
0.0000 - 0.0000i
-0.0000 - 0.0000i
-0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 + 0.0000i
0.0000 - 0.0000i
0.0000 - 0.0000i
-0.0000 - 0.0000i
-0.0000 - 0.0000i
-0.0000 + 0.0000i
Now, my question is, how to transform $z = \sum_{n=0}^{N} \alpha_n\zeta^{n+1}$ into another function as below,
$$z = \alpha_0\zeta + \sum_{n=1}^{N} \dfrac{\alpha_n}{\zeta^n}$$
Note that the series in the second function $n$ is started from $1$.
The goal is to obtain the constant $\alpha_0$ and $\alpha_n$ of the second function as such it will also produce a transformation from an ellipse into a unit circle.
Can anyone help me?
PS. I am thinking of using Fourier series/transform? but I don't really know how to implement them