Complements in Real and Complex Analysis

Among the tools from real analysis to be mastered by a student of machine learning, statistical physics, and thermodynamics, we mention in particular convexity, the main properties of partial derivatives and differentials, and the theory of Lagrange multipliers. Liouville’s theorem on systems of ordinary differential equations plays also an important role. In this chapter, we recall some elementary facts from the theory of functions. Other results include, for instance, the existence of a continuous logarithm for a complex-valued function of a complex variable.