Functions of Multivector Variables

被引：41

作者：

Chappell, James M. ^{[1
]}

Iqbal, Azhar ^{[1
]}

Gunn, Lachlan J. ^{[1
]}

Abbott, Derek ^{[1
]}

机构：

[1] Univ Adelaide, Sch Elect & Elect Engn, Adelaide, SA, Australia

来源：

PLOS ONE | 2015年 / 10卷 / 03期

关键词：

D O I：

10.1371/journal.pone.0116943

中图分类号：

O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];

学科分类号：

07 ; 0710 ; 09 ;

摘要：

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that Cl (R-3) provides a particularly versatile algebraic framework.

引用

页数：21

共 10 条

[1]

[Anonymous], 1995, Clifford Algebras and the Classical Groups

[2]

Boudet R, 2009, SPRINGER SER ATOM OP, V52

[3]

Carrier GF., 2005, Functions of a Complex Variable: Theory and Technique, DOI [10.1137/1.9780898719116, DOI 10.1137/1.9780898719116]

[4]

Greene R.E., 2006, Graduate Studies in Mathematics. Amer. Math. Soc, V40

[5]

Hahn A.J., 1994, Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups

[6]

Hestenes D., 1999, New Foundations for Classical Mechanics, V2

[7]

Janovska D., 2008, MITT MATH GES HAMBUR, V27, P223

[8]

Lam T, 1973, ALGEBRAIC THEORY QUA

[9]

Lounesto P., 2001, CLIFFORD ALGEBRAS SP, V2

[10] Biquaternion (complexified quaternion) roots of-1 [J].

Sangwine, Stephen J. .

ADVANCES IN APPLIED CLIFFORD ALGEBRAS, 2006, 16 (01) :63-68

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