Inaug. Diss.--Bonn. / Bibliography: p. 85-86.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

Includes bibliographical references (p. 62).

Thesis (doctoral)--Universität Jena, 1915. / Vita. Includes bibliographical references.

No description available.

No description available.

It is well known that, as a consequence of the Identity Theorem, we cannot "glue" together two analytic functions to create a new globally analytic function. In this paper we will both introduce and investigate special homeomorphisms, called quasiconformal maps, that are generalizations of the well known conformal maps. We will show that quasiconformal maps make this "gluing," up to conjugation, possible. Quasiconformal maps are a valuable tool in the field of complex dynamics. We will see how quasiconformal maps of infinitesimal circles have an image of an infinitesimal ellipse. Although quasiconformal maps are nice homeomorphisms, they might only be differentiable in the real sense almost everywhere and, surprisingly, complex differentiable nowhere. We shall rely on the work of Lehto and Virtanen as well as Shishikura in exploring these interesting complex valued functions. / Department of Mathematical Sciences

Honors thesis (B.A.)-Ithaca College Dept. of Mathematics, 2007. / "May 2007." Includes bibliographical references (leaf 24). Also available in print form in the Ithaca College Archives.