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Four Dimensional Graphs Of Complex Functions, Malcom Lee Murrill

Master's Theses

Complex functions of a single complex variable involve four unknowns, two independent and two dependent variables, and thus cannot be adequately represented in two- or three- dimensional space. Various geometric constructions in both two and three dimensions have been devised in the past, however, in attempts to illuminate complex function theory. The standard and most useful, of these representations is that developed by Gauss and Riemann employing two complex planes simultanesously. These show the correspondence between a particular curve or region in the object plane and its image, as mapped by a given transformation, in the image plane.

A Historical Survey Of Methods Of Solving Cubic Equations, Minna Burgess Connor

Master's Theses

It has been said that the labor-saving devices ot this modern age have been made possible by the untiring efforts of lazy men. While working with cubic equations, solving them according to the standard methods appearing in modern text-books on the theory of equations, it became apparent, that in many cases, the finding of solutions was a long and tedious process involving numerical calculations into which numerous errors could creep. Confessing to laziness, and having been told at an impressionable age that "any fool can do it the hard way but it takes a genius to find the easy …

The Existence Theorem Of Ordinary Differential Equations, Harris J. Dark

Master's Theses

There are a great many devices for solving differential equations of certain special forms. But there is a large number of classes of differential equations that are not included in these special forms and cannot be integrated by quadratures or other purely elementary methods. When mathematicians were forced to abandon their cherished hope of finding a method for expressing the solution of every differential equation in terms of a finite number of known functions or their integrals they turned their attention to the question of whether a differential equation in general had a solution at all, and, if so, of …

Gauss' Hypergeometric Equation, William R. Smith

Master's Theses

As early as the seventeenth century the English mathematician, john Wallis (1616-1703), used the term "hypergeometric" to describe a series which he was studying. This series, ∑(a)(a+b)(a+2b)…(a+n-1b), is quite different from the usual geometric series, hence the term, "hyper" (=above) plus "geometric," was used to signify that the series was of greater complexity than the geometric series. Wallis did not consider his series a power series or a function of x.

In 1769 this series received a remarkable development at the hands of Loonhard Euler who, following the example of Wallis, applied the word "hypergeometric" to it. He observed that …