Differentiability and analyticity of functions in linear algebras

Trampus, A.

doi:10.1215/s0012-7094-60-02741-1

Cited by 5 publications

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“…Our work is probably more inline with that of Wagner to be honest. Ward's analytic functions for noncommutative algebras have been studied by Trampus [45] and Rineheart [37].…”

Section: Historymentioning

confidence: 99%

Introduction to $\mathcal{A}$-Calculus

Cook¹

2017

Preprint

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Let A denote an n-dimensional associative algebra over R. This paper gives an introductory exposition of calculus over A. An A-differentiable function f : A → A is one for which the differential is right-A-linear. The basis-dependent correspondence between right-A-linear maps and the regular representation of real matrices is discussed in detail. The requirement that the Jacobian matrix of a function fall in the regular representation of A gives n 2 −n generalized A-CR equations. In contrast, some authors use a deleted-difference quotient to describe differentiability over an algebra. We compare these concepts of differentiability over an algebra and prove they are equivalent in the semisimple commutative case. We also show how difference quotients are ill-equipt to study calculus over a nilpotent algebra.Cauchy Riemann equations are elegantly captured by the Wirtinger calculus as ∂f ∂ z = 0. We generalize this to any commutative unital associative algebra. Instead of one conjugate, we need n − 1 conjugate variables. Our construction modifies that given by Alvarez-Parrilla, Frías-Armenta, López-González and Yee-Romero in [16].We derive Taylor's Theorem over an algebra. Following Wagner, we show how Generalized Laplace equations are naturally seen from the multiplication table of an algebra. We show how d'Alembert's solution to the wave-equation can be derived from the function theory of an appropriate algebra. The inverse problem of A-calculus is introduced and we show how the Tableau for an A-differentiable function has a rather special form. The integral over an algebra is also studied. We find the usual elementary topological results about closed and exact forms generalize nicely to our integral. Generalizations of the First and Second Fundamental Theorems of Calculus are given. This paper should be accessible to undergraduates with a firm foundation in linear algebra and calculus. Some background in complex analysis would also be helpful.

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“…Our work is probably more inline with that of Wagner to be honest. Ward's analytic functions for noncommutative algebras have been studied by Trampus [45] and Rineheart [37].…”

Section: Historymentioning

confidence: 99%