# Exponentials of general multivector in 3D Clifford algebras

@article{Dargys2021ExponentialsOG, title={Exponentials of general multivector in 3D Clifford algebras}, author={Adolfas Dargys and Artūras Acus}, journal={Nonlinear Analysis: Modelling and Control}, year={2021} }

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image…

## 4 Citations

### Coordinate-free exponentials of general multivector in Cl(p,q) algebras for p+q=3

- MathematicsMathematical Methods in the Applied Sciences
- 2021

Closed form expressions in real Clifford geometric algebras Cl(0,3),
Cl(3,0), Cl(1,2), and Cl(2,1) are presented in a coordinate-free form
for exponential function when the exponent is a general…

### The characteristic polynomial in calculation of exponential and elementary functions in Clifford algebras

- Mathematics
- 2022

. Formulas to calculate multivector exponentials in a base-free representation and in a orthonormal basis are presented for an arbitrary Cliﬀord geometric algebra Cl p,q . The formulas are based on…

### Current survey of Clifford geometric algebra applications

- MathematicsMathematical Methods in the Applied Sciences
- 2022

We extensively survey applications of Cliﬀord Geometric algebra in recent years (mainly 2019–2022). This includes engineering, electric engineering, optical ﬁbers, signal-, image- and video…

### Logarithm of multivector in real 3D Clifford algebras

- Mathematics
- 2023

Closed form expressions for a logarithm of general multivector (MV) in base-free form in real geometric algebras (GAs) Cl(p,q) are presented for all n=p+q=3. In contrast to logarithm of complex…

## 26 References

### Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra

- Mathematics
- 2011

This paper presents equations for the inverse of a Clifford number in Clifford algebras of up to five dimensions. In presenting these, there are also presented formulas for the determinant and…

### Multivector and multivector matrix inverses in real Clifford algebras

- MathematicsAppl. Math. Comput.
- 2017

### A New Approach to Differential Geometry using Clifford's Geometric Algebra

- Mathematics
- 2011

Preface.- Introduction.- Clifford Algebra in Euclidean 3-Space.- Clifford Algebra in Minkowski 4-Space.- Clifford Algebra in Flat n-Space.- Curved Spaces.- The Gauss-Bonnet Formula.- Non-Euclidean…

### On determinant, other characteristic polynomial coefficients, and inverses in Clifford algebras of arbitrary dimension

- MathematicsArXiv
- 2020

This paper presents the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras.

### Characteristic Polynomials in Clifford Algebras and in More General Algebras

- MathematicsAdvances in Applied Clifford Algebras
- 2019

This article combines two independent theories: firstly, the algorithm of Faddeev–Leverrier which calculates characteristic polynomials of matrices; secondly, the Descent Theory which, in particular,…

### Exponential factorization of multivectors in C l ( p , q ), p + q <3

- Mathematics, Computer Science
- 2020

In this paper we consider general multivector elements of Clifford algebras Cl(p,q), p+q <3, and study multivector factorization into products of exponentials and idempotents, where the exponents are…

### Exponential Factorization and Polar Decomposition of Multivectors in $Cl(p,q)$, $p+q \leq 3$

- Mathematics, Computer Science
- 2019

In this paper we consider general multivector elements of Clifford algebras $Cl(p,q)$, $n=p+q \leq 3$, and study multivector equivalents of polar decompositions and factorization into products of…

### Functions of Multivector Variables

- MathematicsPloS one
- 2015

A number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions are detailed, finding one key relationship that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation.

### Geometric Algebra for Physicists

- Mathematics, Physics
- 2003

Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with…