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The elementary function theory of an hypercomplex variable and the theory of conformal mapping in the hyperbolic plane Fox, Geoffrey Eric Norman

Abstract

The present thesis is based on a paper by Bencivenga. In this paper the author develops a theory of function for the dual and bireal variables. He constructs the "retto" and "hyperbolic" planes for the geometric representation of the dual and bireal variables, respectively, and establishes a type of conformal mapping of these planes into themselves by means of differentiable functions of the variable. Further, in each of these planes he proves the analogue for the Cauchy integral theorem of the complex plane. Finally he shows that functions of the dual and bireal variable which possess all derivatives at a given point of the plane may be expanded in a Taylor series about that point. In the first chapter we give a summary of this paper. Bencivenga’s dual and bireal number systems, and also the complex number system, are two-dimensional cases of the ɳ - dimensional associative, commutative linear algebra with unit element. In chapter II we generalize Bencivenga's function theory to functions over the above mentioned linear. An important class of results from the theory of functions of a complex variable are not generalizable, since they depend on the field properties peculiar to the complex algebra. In chapter III we undertake a detailed study of the hyperbolic plane with particular reference to the conformal properties of differentiable functions of the bireal variable, as a special case of conformal transformation of the hyperbolic plane, we study the bilinear transformation. We find that the rectangular hyperbola is the geometrical form which is invariant under this transformation of the hyperbolic plane. Singularities play a larger role in this theory than in the case of the analgous transformation theory of the complex plane.

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