## Complex Representations of Functions

When one considers the representation of analog signals defined over an infinite interval and containing a continuum of frequencies, one will see that Fourier series sums become integrals of complex functions and so do the Fourier coefficients. This chapter reviews some facts about complex numbers and then introduce complex functions. This will lead us to the calculus of functions of a complex variable, including the differentiation and integration complex functions. Complex numbers were first introduced in order to solve some simple problems. The chapter explores complex functions and the calculus of complex functions. It introduces functions of a complex variable. The chapter aims to establish when functions are differentiable as complex functions, or holomorphic. It investigates the computation of complex path integrals.