This chapter introduces the fundamentals of the Fourier series, beginning with a historical look at Joseph Fourier’s work on heat distribution using trigonometric series, which paved the way for modern Fourier analysis. It covers essential mathematical concepts, including the properties of sinusoids, orthogonal functions, and the roles of odd and even functions in decomposing complex periodic signals. The chapter also discusses function series expansions, enabling the representation of complex functions as infinite sums of simpler terms. Key principles of complex analysis are introduced, with a focus on essential differentiation and integration formulas for complex variables. These mathematical foundations are established to support a deeper understanding of the Fourier series and its applications.