Calculation of vectorial derivatives for functions of a quaternion variable and their properties

Abstract

AbstractVarious attempts have been made in defining the derivative of a quaternionic function due to the noncommutativity of the product over quaternions. We observe that the difference in the left and right operations caused by the noncommutativity of the quaternion product is determined by the vector part of the quaternion. In this paper, we propose a corresponding derivative to replace the derivative of a quaternion-valued function of a quaternionic variable using the component terms of a quaternion. Further, the analogous constant, product, and quotient rules for the proposed calculations are given. Application of the proposed derivatives is provided to compute the derivatives of elementary functions. Several illustrations are also presented.