The fact, namely, that the analytic functions are a very limited and special class, with the additional fact that there seems to be no reason a priori why many of the theorems concerning analytic functions cannot be extended ...

In this paper we propose to study the behavior of series of complex numbers, or of vectors in two dimensions; and to generalize this study to the case of vectors in n dimensions. The particular properties to be studied are ...

In the classic memoirs of Sturm and Liouville, two classes of theorems are found concerning sets of orthogonal functions. The first deal with the number of sign-changes in [phi]3, and the second with the number of sign-changes ...

In this present paper we will develop some theorems concerning the degree of convergence of certain series, in particular a Fourier's series, a Legendre's series, and a series of Bessel's functions. Before proceeding ...

Realizing that the Galois theory of algebraic equations as commonly presented seems artificial, abstract, and intricate, we have been led in the following paper to attempt to present in a clear, tangible fashion the general, ...

In Part I of this paper, I have dealt with only well-known properties of functions - treating them from the graphic standpoint entirely and referring the reader, to the best authorities I could find, for the Algebraic ...

The concept of pointwise discontinuity is a fairly recent one in mathematics. Originally introduced as a convenient term in the study of integration, it has quite outgrown its former sphere of usefulness and has had an ...

Essentially, a group is an associative field, in which the inverse combinations are uniquely possible. This is a concise statement of the classical definition of a group. The conditions which it connotes will be used here ...

This thesis gives some of the more important tests for the convergence of an infinite series; also the conditions that must be fulfilled in order that certain operations and transformations may be applied to an infinite series.

A differential equation may be formed from all algebraic equations by the elimination of the arbitrary constants between the latter and its derivatives. The number of derivations being equal to the number of arbitrary ...

1. Notations. Suppose that we have given a plan curve C, and a point O, not lying in the plane of the curve. If we draw vectors from point O, one to every point of the given curve, we produce a conical surface, the elements ...

In this dissertation I propose to give some of the theory and develop some of the important formulas upon which Minimum Surfaces are based. In order to proceed with the development of Minimum Surfaces, it will be necessary ...

We have shown that the pseudosphere is applicable to itself in an infinity of ways. Therefore these surfaces that are applicable to it can, after they are folded on the pseudosphere, be made to pass through the same ...

The developments of the last twenty years in the theory of sets of points and in the applications of this theory to the theory of functions of real variables, besides leading to a tremendous extension of the ordinary theory ...

The subject of this paper is the study of the four problems of: (1) integration of a series term by term; (2) differentiation of a series term by term; (3) reverse of the order of integration in a double integral; (4) ...

The problem discussed in these pages is that of the motion of a billiard ball when struck by a horizontal cue. This is a special case of the motion of a sphere on a rough horizontal plane.

We shall define an infinite series as a succession of series formed after sum definite law. Most generally the series are actual numbers or are at least regarded as constraints, and we are concerned with their sum. There ...

The subject matter of this paper was suggested by the belief that a treatment of irrational numbers, from the stand point of the "cut" number, has certain points of superiority over the common treatment from the standpoint ...

The interest attaching to n-dimensional geometry comes chiefly from two sources, first the light thrown upon analysis by a geometric interpretation of its results when more than three variables are involved, and second, ...

In speaking of the icosahedron and other regular solids in the following work we shall include not only the space construction but also the sphere surface upon which the corners, edges and faces of the solids may be ...