08. Scientific Journals

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Functional Equations: Review of Solution Methods and Applications in Real and Complex Analysis
(Suleyman Demirel University, 2011) Muammer Gul
This article provides an overview of functional equations, one of the oldest and most fundamental topics in mathematical analysis. It discusses classical types of equations, including Cauchy’s additive and multiplicative equations, and highlights their historical development through the works of D’Alembert, Poisson, Cauchy, and Lobachevsky. Several well-known equations, such as those related to the parallelogram law, hyperbolic functions, and non-Euclidean geometry, are illustrated as examples of how functional equations arise naturally in mathematics. The paper also explores the existence of discontinuous solutions, the role of continuity and boundedness assumptions, and how these conditions restrict solutions to linear or exponential forms. Finally, it emphasizes that many functional equations characterize broad classes of functions—such as periodic or symmetric functions—and outlines general strategies for solving them.