III.—A Series Formula for the Roots of Algebraic and Transcendental Equations

Abstract

The object of this paper is to communicate a general formula for elementary symmetric functions of any assigned degree of a given number of the roots, in ascending order of magnitude, of an algebraic or transcendental equation with complex coefficients. In virtue of the simple relations that exist between these symmetric functions, the formula gives a literal expression in terms of infinite series for all the roots of an equation. In practice it is generally desirable to transform the equation first into one with roots widely separated in value (e.g. by the root-squaring process) in order to make the convergence of the series sufficiently rapid for satisfactory numerical computation.