# Inverse of a function with summation

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.

• nobody is interested to reply to my question? :) – BeeTiau Nov 13 '18 at 18:02

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 $$z$$.
• 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. – BeeTiau Nov 15 '18 at 16:19