Quaternionic Analysis

Abstract: In this thesis, we study Quaternionic Analysis, which is the most natural and close generalization of complex analysis. Quaternion Analysis conserves many of its important features by key reference A. Sudbery. Quaternions are a non-commutative multiplication system in which all the other field hypotheses are valid, so the investigation of their properties and structure became the basis of this study. The first chapter contains a brief history of what led to the discovery of quaternions and their construction as a fourdimensional. Chapter two develops quaternionic algebra which is become a common part of mathematics and physics culture. In the third chapter, we present some theorems on therepresentations of quaternions utilising regularity and Cauchy-Riemann-Fueter. Quaternion derivatives in the mathematical literature are typically defined only for analytic (regular) functions. Moreover, this chapter shows how regular functions can be constructed from harmonic functions. The fourth and last chapter summarises the weaknesses and strengths of this thesis and provides suggestions for further study.