The Right Setting of the Quaternion Calculus

1987

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Bulletin des Sciences Mathématiques, 2003

We investigate differentiability of functions defined on regions of the real quaternion field and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as criteria for functions of a quternion variable to be analytic. In particular, the quaternionic exponential and logarithmic functions are being considered. Main results include quaternion versions of Hurwitz' theorem, Mittag-Leffler's theorem and Weierstrass' theorem.

Provides an introduction to the concept and derivation of the Quaternions. Then derives several basic algebraic relationships involving quaternions, culminating with how quaternions can be used to describe 3d rotations. Then discusses the computational advantages that quaternions have over other 3d rotation methods, and provides an example of a classical mechanics rotation problem being solved with quaternions. Then discusses Frobenius's Theorem and the potential physical implication that it has. Overall aim is to provide a good educational resource for the intuition of quaternions and their applications.

Recent innovations in the differential calculus for functions of noncommuting variables, beginning with a quaternionic variable, are now extended to consider some integration. *

2000

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential equations with constant coefficients. We overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. The resolution of the complex linear Schrodinger equation, in presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.

This is a compilation of quaternionic number systems, quaternionic function theory, quaternionic Hilbert spaces and Gelfand triples. The difference between quaternionic differential calculus and Maxwell based differential calculus is explained.

Computers & Mathematics With Applications, 2004

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential equations with constant coefficients. We overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. The resolution of the complex linear Schrödinger equation, in presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.

Trends in Mathematics, 2013

The present paper is aimed at proving necessary and sufficient conditions on the quaternionic-valued coefficients of a first-order linear operator to be associated to the generalized Cauchy-Riemann operator in quarternionic analysis and explicitly we give the description of all its nontrivial first-order symmetries.

We draw the conclusions from the earlier presented quaternionic generalization of Cauchy-Riemann's equations. The general expressions for constituents of -holomorphic functions as well as the relations between them are deduced. The symmetry properties of constituents of -holomorphic functions and their derivatives of all orders are proved. For full derivatives it is a consequence of uniting the left and right derivatives within the framework of the developed theory. Some -holomorphic generalizations of ℂ − holomorphic functions are discussed in detail to demonstrate particularities of constructing H-holomorphic functions. The power functions are considered in detail.

Journal of Geometry and Physics, 2010

In the paper [F. Colombo, I. Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009) 601-627] the authors treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute. This functional calculus is the quaternionic version of the classical Riesz-Dunford functional calculus. When considering quaternionic operators it is natural to also consider the case of left linear operators. Furthermore, one can use left or right slice regular functions to construct a functional calculus for right (or left) linear operators. In this paper we discuss these possibilities, showing that, in all the cases, we can associate to an operator two so-called S-resolvent operators but their interpretation depends on whether we are considering a left or a right linear operator. Also the S-resolvent equations for right or left closed operators do not have the same interpretation. Moreover, we study the bounded perturbations of both the S-resolvent operators.