Study Of Some Problems On Quaternion Algebras And Analysis

Quaternion is a non-commutative and associative division algebra and it is extended to the field of complex numbers. In recent twenty years, the quaternion is widely applied to rigid mechanics, graph theory of computer, robot technology, artificial satellite attitude control ,and so on. In this paper, we study some problems of quaternion algebras and analysis in terms of new ways, the main results are as follows:1. The algebraic structure and some properties of quaternion are obtained. Such as various kinds of representation format of quaternion, rule of operation, the convergence of quaternion normed spaces, the continuity of quaternion inner spaces, Cauchy-Schwartz inequality, parallelogram formula and polarization identity, etc.2. Two types of definitions of quaternion analytical function are analytic,we obtain two necessary and sufficient conditions of quaternion regular function. At the same time, a sufficient condition of quaternion harmonic function are obtained, too.3. We studied quaternion regular function by exterior differential. Two necessary and sufficient conditions of quaternion left (or right) regular function are obtained. and we proved Cauchy theorem and Cauchy integral formula over the quaternion field.