On The Conformal Representation Of Plane Curves Particularly For The Cases P PDF Download

Excerpt from On the Conformal Representation of Plane Curves Particularly for the Cases P = 4, 5, and 6 In order to do this we must classify surfaces according to their deficiencies and treat each class separately. In the case p=0 there are no cuts on the surface and no integrals of the first kind. Integrals of the second kind exist on all surfaces. Let us select one of these, w, which has a single algebraic infinity. The function w being of weight one assumes one and only one value corresponding to each point of the n-leaved surface F (x,y) = 0. These values, real and complex, may be represented by the points in a plane by the ordinary representation of complex numbers. The given n-leaved surface can then be conformally represented upon a plane by means of the real and complex values assumed by w. If p = 1 two cuts are required to make the surface simply connected. We know that on any surface of deficiency p, there exist p linearly independent integrals of the first kind. Here then there exists only one which we will call u. Let the periods across the cuts be w1 and w2 If u has the value uo at a given point, for all the region around containing no branch point the u will vary continuously, and since u can be nowhere infinite, and since its values may be represented by points of a plane just as any complex number is represented, these points must all be within a parallelogram whose boundaries are determined by the limits of the values of the real and imaginary parts of u as it varies over the surface, never crossing a boundary. If we seek then to represent our entire surface by means of the integral u which has an infinite number of values we get corresponding to a given point of Fn an infinite number of homologous points in similar parallelograms. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.