Zeros of Partial Fractions 363 Zero - Free Regions of Linear Partial Fractions

when the a/s are complex constants. The case of real a/s has already been adequately treated by Gauss, Lucas, Jensen, and Bôcher in connection with their study of the derivative of a polynomial and the jacobian of two binary forms. In terms of $, their results may be stated as follows. (a) If all the ay's have the same sign, there are no zeros of $ outside of any convex polygon enclosing the points a,-. If all the dj, in addition, lie on the line-segment AB, there are no zeros of except at points on A B $. (b) If all the a/s have the same sign, and the a/s are either real or in conjugate imaginary pairs, there are no imaginary zeros of $ at points outside of all the Jensen circles.|| In any Jensen circle containing k a / s , and not overlapping any other Jensen circle, there are at least k — 1 and at most k + 1 roots of <Mf