Suppose that I have the following function: $$z(\zeta)=\sum_{k=0}^{n}m_k\zeta^{1-k}$$
How do I get the inverse of that function? i.e. I want to express $\zeta(z)$?
In my case, $10\leq n \leq 20$.
In my case, $z$ cannot be zero and the constants $m_k$ is not arbitrary but rather obtained from another process. This is actually a conformal mapping function that can map any shape into a unit circle, thus $z$ cannot be zero.
Is it possible to have a general rule to defined $\zeta(z)$? I can then translate them into Matlab, for instance.
In general your function will not have an inverse function.
Consider for example $$z(\zeta):=\zeta-3+\frac{2}{\zeta}$$ which has the property $$z(1)=z(2)=0.$$ Thus, it's not one-to-one and you can't define an inverse function on the whole range of $z$.