Analytic Properties of the Quaternion Function

Youcai Xue

doi:10.5539/mas.v4n10p62

Abstract

Many properties of complex functions are pretty difficult to be generalized in the field of quaternion function, as the commutative law of multiplication fails in the latter. The derivative of quaternion function is defined in this paper. Moreover, by the similar method in judging the analytic property of complex function, Cauchy-Riemann equation is used to determine the analytic property of quaternion function. Furthermore, several concrete examples are discussed, and certain errors in (P. W. Yang. 2009) are pointed out as well.

Highlights

The quaternion function is an important aspect of functions theory
We define the derivative of quaternion function and give a different definition of its analytic property
Let x x1, x2, x3, x4 be some element in R4, such that U : D Q ; w f (q) u1 u2i u3 j u4k, where ui (x1, x2, x3, x4 ), i 1, 2, 3, 4 are real functions defined on D

Summary

Introduction

The quaternion function is an important aspect of functions theory. With a similar method of Cauchy-Riemann equation in complex functions, 2009) defines the analytic of quaternion function directly. We define the derivative of quaternion function and give a different definition of its analytic property. Let x x1, x2 , x3, x4 be some element in R4 , such that U : D Q ; w f (q) u1 u2i u3 j u4k , where ui (x1, x2 , x3, x4 ), i 1, 2, 3, 4 are real functions defined on D .

Results

Conclusion