can someone help me how to find the inverse the following function?
$$z(\zeta)=\frac{1}{\zeta}+m_1\zeta+m_2\zeta^2+m_3\zeta^3$$
In my case, $z$ is a complex number and cannot be zero. And $m_k$ is a constant.
How do I get the inverse of that function? i.e. I want to express $\zeta(z)$? IS it possible to have an explicit formulation? Or, for a given $z$ can we solved $\zeta(z)$ numerically?
Example
Consider the following function: $$z(\zeta)=\frac{1}{\zeta}$$
In my understanding, the inverse of the above function is $$\zeta(z)=\frac{1}{z}$$
Your help is highly appreciated.
The answer of this question has be given by @nathan.j.mcdougall above. It is essentially solving numerically the following equation
$$m_3\zeta^4+m_2\zeta^3+m_1\zeta^2−z\zeta+1=0$$