\[ \newcommand{\e}{\mathrm{e}} \DeclareMathOperator{\id}{id} \]

Let \(f:U\to \mathbb{R}\) be real analytic and \(f'(x_0)\neq 0\) for some \(x_0\in U\). Then there is a neighbourhood \(V\) of \(x_0\) and a real analytic function \(g\colon V\to \mathbb{R}\) such that

\begin{equation*} f\circ g=g\circ f=\id. \end{equation*}

[1, Proposition 1.5.3]

Proof
We can prove this statement applying the inverse function theorem and the characterization (0x675c0899) .

References Link to heading

  1. S. Krantz and H. Parks, A Primer of Real Analytic Functions. Boston, MA: Birkhäuser Boston, 2002. doi:10.1007/978-0-8176-8134-0