This text is a detailed mathematical treatise on linear and nonlinear integral equations, focusing on the theory of eigenvalues and eigenfunctions. It begins with an introduction to Fredholm's solution of the inhomogeneous linear integral equation, which leads to the concept of eigenvalues and eigenfunctions. The work then explores the development of arbitrary functions in terms of eigenfunctions, using Hilbert's approach involving orthogonal functions and quadratic forms. It discusses the existence of eigenvalues, the development of functions in series of eigenfunctions, and the implications for both symmetric and asymmetric kernels. The text also addresses the approximation of functions using eigenfunctions, the convergence of series, and the properties of eigenvalues and eigenfunctions in different contexts. It includes proofs of fundamental theorems, the behavior of eigenvalues under various conditions, and the extension of results to non-continuous kernels. The treatise concludes with a discussion of the implications for nonlinear integral equations and the resolution of such equations through eigenfunction expansions. The work is comprehensive, covering both theoretical foundations and practical applications in the theory of integral equations.This text is a detailed mathematical treatise on linear and nonlinear integral equations, focusing on the theory of eigenvalues and eigenfunctions. It begins with an introduction to Fredholm's solution of the inhomogeneous linear integral equation, which leads to the concept of eigenvalues and eigenfunctions. The work then explores the development of arbitrary functions in terms of eigenfunctions, using Hilbert's approach involving orthogonal functions and quadratic forms. It discusses the existence of eigenvalues, the development of functions in series of eigenfunctions, and the implications for both symmetric and asymmetric kernels. The text also addresses the approximation of functions using eigenfunctions, the convergence of series, and the properties of eigenvalues and eigenfunctions in different contexts. It includes proofs of fundamental theorems, the behavior of eigenvalues under various conditions, and the extension of results to non-continuous kernels. The treatise concludes with a discussion of the implications for nonlinear integral equations and the resolution of such equations through eigenfunction expansions. The work is comprehensive, covering both theoretical foundations and practical applications in the theory of integral equations.