ABSTRACT
This chapter presents the fundamental properties of characteristic functions. In terms of characteristic functions, the complex operation is replaced by a simple multiplication of characteristic functions. Two limit theorems, direct and converse, are most important for applying characteristic functions in deriving asymptotic formulas in probability theory. These theorems establish that the correspondence existing between distribution functions and characteristic functions is not only one-to-one but also continuous. It is important that the converse theorem also holds: a distribution function is uniquely determined by its characteristic function. The mathematical expectation and variance can be very simply expressed in terms of the derivatives of the logarithm of the characteristic function. If a distribution function is symmetric, then its characteristic function is real. The chapter provides an exhaustive description of the class of characteristic functions. The fundamental theorem was simultaneously proved by A. Ya. Khinchine and Bochner and first published by Bochner.
