## 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 $$\begin{cases} A_0(x)=x \\ A_{n+1}(x)=\frac{A_n'(x)}{f'(x)}\end{cases}$$ It is evident that for some classes of functions starting from some point $A_k(x)$ becomes zero and thus the inverse function can be expressed in closed form. For example, the expression has … Read more