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Functions of multivector variables.

James M Chappell1, Azhar Iqbal1, Lachlan J Gunn1

  • 1School of Electrical and Electronic Engineering, University of Adelaide, Adelaide, South Australia, Australia.

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Summary
This summary is machine-generated.

This study extends elementary functions to Clifford multivectors, revealing new connections between complex numbers, quaternions, and vectors. A key finding shows complex numbers raised to vector powers yield quaternions.

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Area of Science:

  • Mathematics
  • Algebraic Geometry
  • Theoretical Physics

Background:

  • Elementary functions are well-defined over real and complex numbers.
  • These functions can be generalized to the skew field of quaternions.
  • Clifford algebra offers a framework for unifying various number systems.

Purpose of the Study:

  • To extend elementary functions to Clifford multivectors in two and three dimensions.
  • To explore inter-relationships between complex numbers, quaternions, and Cartesian vectors within Clifford algebra.
  • To develop a unified framework for multivector operations.

Main Methods:

  • Generalization of elementary functions to the Clifford algebra framework.
  • Algebraic manipulation of multivector expressions.
  • Comparative analysis of functions across different dimensional Clifford algebras.

Main Results:

  • Demonstration of new inter-relationships between complex numbers, quaternions, and vectors within multivectors.
  • Discovery that a complex number raised to a vector power results in a quaternion.
  • Development of a single formula for the square root, amplitude, and inverse of a multivector in 1D, 2D, and 3D.

Conclusions:

  • Clifford multivectors provide a unified algebraic system for complex numbers, quaternions, and vectors.
  • The Clifford algebra Cl(R(3)) offers a particularly versatile framework for these extended functions.
  • The findings establish novel connections and unified operational capabilities within advanced algebraic systems.