The Lagrange inversion theorem is as follows: Let $f$ be a function which is holomorphic in a neighborhood of a with $f(a) = 0, f'(a) \neq 0$. We know that the functional inverse of f is well-defined in a neighborhood of 0. Moreover, the Taylor series of f is given by $ f^{-1}(u) = a + \sum_{k=1}^\infty \frac{1}{k} \left(res_{z = a} \frac{1}{f(z)^k} \right)u^k$ with $res_{z=a}$ denoting the Cauchy residual at $a$.
Now, assume that $f(b) = 1$ holds. Considering $g(u) := a + \sum_{k=1}^\infty \frac{1}{k} \left(res_{z = a} \frac{1}{f(z)^k} \right)u^k$ as a formal power series, can we conclude that $b = g(1)$ holds?
On Wikipedia (and some books including the reference given in Wikipedia) it says that $g(f(z)) = z$ holds true, i.e. $b = g(1)$ would be true and we can at least find the inverse of $f$ as a formal power series. However, from my point of view $g(f(z)) = z$ should only hold in a neighborhood of a. What is correct?
Thanks in advance!