A Geometrical Problem Connected with the Continuation of a Power-Series

Abstract

GIVEN a power-series P (x) with only one singular point A on the circumference of its circle of convergence. Denote this circle by (Cl, its center by M. Let us suppose that all the other singular points of the analytic function defined by the element P(x) are so situated as not to interfere with the continuations of P(x) which are to be considered. The following purely geometrical question arises: How are the intermediate circles to be chosen in order to arrive again at a circle with center -ll, by a minimum number of continuations around the point A? To answer this question we consider the total area covered by all direct (first) continuations of P(x). The boundary of this area is the envelope of the circles passing through A whose centers lie on the circumference of C1. This envelope is a cardioid 02. The boundary of the area covered by all the second continuations is the envelope C3 of all circles through A whose centers lie on C2. Continuing this process, we obtain a series of curves C2, C03, . * (n, where U, is the boundary of the area covered by all (n 1)st continuations. Counting radii vectores p from A and polar angles 9 from AM, and taking the radius AM= 1, the equations of the successive curves appear in