Date of Award

Summer 1948

Degree Type

Thesis - Restricted

Degree Name

Master of Science (MS)

Department

Mathematics

Abstract

A point in the complex plane which is subjected to a linear fractional transformation may be thought of as moving along a continuous path from its initial position to its final position under the transformation. A detailed algebraic as well as geometric description of this path, or track, is not to be found in the standard references on functions of a complex variable which were available to the writer. The first three chapters of this theses provide (1) a brief and very elementary treatment of the theory of these tracks, and (2) two particular examples of tracks. It will be seen that the problem of describing the track does not rigorously involve the idea of motion but resolves itself to the problem of finding a system of one-parameter families of transformations such that the transformed points of these families form a continuous locus from the initial position of z to its final position under a given transformation. The problem of classifying the linear fractional transformations is treated slightly differently by Forsyth, Ford, and Townsend and is completely omitted by several other authors. The purpose of the fourth chapter is to correlate Townsend's treatment of the subject with Forsyth's. In order (1) to provide a way of quickly locating secondary sources for various areas of this theory, and (2) to provide a very brief comparison and contrast of methods of treatment of linear fractional transformations, the bibliography is rather carefully annotated.

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