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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.

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  • $\begingroup$ nobody is interested to reply to my question? :) $\endgroup$ – BeeTiau Nov 13 '18 at 18:02
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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 $z$.

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  • $\begingroup$ 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. $\endgroup$ – BeeTiau Nov 15 '18 at 16:19
  • $\begingroup$ Then you should add these additional condition to your question! $\endgroup$ – weee Nov 20 '18 at 9:05
  • $\begingroup$ Done. I have added the condition into my question. ^_^ $\endgroup$ – BeeTiau Nov 20 '18 at 12:44

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