In this brief chapter, we cover the foundational concepts of real analytic functions needed for Part II, especially for proving the Cauchy–Kovalevskaya and Holmgren theorems for the Cauchy problem with analytic coefficients, as well as the John Theorem on continuous dependence estimates. The key idea is the power series in multiple variables, an extension of the single-variable case. We also introduce majorant functions and series, essential for the Cauchy–Kovalevskaya Theorem and useful for showing that analytic functions are closed under sum, multiplication, and composition.