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### For which classes of functions this inverse function formula gives a closed form expression?

Lets consider this method of finding inverse function:
$$f^{-1}(x) = \sum_{k=0}^\infty A_k(x) \frac{(x-f(x))^k}{k!}$$
where coefficients $A_k(x)$ recursively defined as
$$\begin{cases} A_0(x)=x \\ ...