Abstract: We will state and prove the so-called Poincaré-Siegel theorem for the existence of a local holomorphic change of variables that conjugates a holomorphic map to its linear part in a neighborhood of a fixed point of the map. The proof is perhaps the simplest illustration of the Newton method - the main tool in the proof of the celebrated Kolmogorov-Arnold-Moser theorem in dynamical systems. No background in complex analysis or dynamical systems is required to understand the proof.