RESUMENEl objetivo principal de este artculo es dar una vision general sobre funciones elemen-tales en el contexto de analisis cuaternionico. De nimos algunas de sus propiedadesmas comunes, que como en los casos reales y complejos, seran familiares para el lec-tor. Esto lleva a la consideracion de las funciones de valor-cuaternionico dependiendode una variable cuaternionica, esto es, funciones las cuales de entrada y salida soncuaterniones.Keywords and Phrases: Quaternionic analysis, elementary functions.2010 AMS Mathematics Subject Classi cation: 30G35, 30A10. 1 Introduction As is well known, quaternionic analysis generalizes the theory of holomorphic functions of one com-plex variable and also provides the foundations to re ne the theory of harmonic functions in higherdimensions. Methods of quaternionic analysis in combination with other classical and modernanalytical methods (such as harmonic analysis, variational methods, and nite di erence methods)have been playing an increasingly active part in the treatment of problems, mainly in mathemati-cal physics, which involve the treatment of elementary functions. Basic results were independentlydiscovered and rediscovered by many people, among others: Sche ers (1893), Dixon (1904), Lanc-zos (1919), Moisil-Teodorescu (1931), Melijhzon (1948), Iftimie (1965), Hestenes (1968), Delanghe(1970), and Sudbery (1979), Brackx, Delanghe and Sommen (1982), Gurleb eck and W. Sproˇig(1989). Meanwhile quaternionic analysis has became a well established branch in mathematicsand greatly successful in many di erent directions. Navigation, computer vision, robotics, signaland image processing, or ecient description of classical mechanics and electrical engineering areexamples of elds where quaternions are used nowadays. A survey within the scope of quaternionicanalysis and its applications is given in [10], and references therein.The organization of this paper is as follows. Section 2 begins with a review of some de nitionsand basic properties of quaternionic analysis. We proceed in Section 3 to study the quaternionexponential and logarithmic functions. Although quaternion multiplication is not commutative,many formal properties of the complex exponential and logarithmic functions can be generalizedwithin this framework. In the remaining sections the quaternion trigonometric, hyperbolic, andtheir inverse functions are covered. A brief discussion on the notions of multiple-valued functionsand branches is also presented. There is no attempt to cover everything related to elementaryquaternion functions and review the historical development of quaternions. General informationis contained in the books [6, 7].