The Imaginary Roots of Equations

Abstract

IN discussing the stability of an oscillating system, it is often necessary to know whether the period-equation has any root the real part of which is positive. We proceed to show how to find out the number of such roots. Let the equation f(z) = 0 be of degree n and let f(iy) = u + iv where u and v are real. If v is of higher degree than u put f₁(y) = v, f₂(y) = u, otherwise put f₁(y) = u, f₂(y)= v. Go through the operation of finding the G.C.M. of f₁ and f₂ with the difference that the sign of each remainder is changed before it is recorded or used as a dividend (just as in getting Sturm's Functions), and let the remainders (with changed signs) be f₃(y), f_m(y). Let the number of changes of sign in the sequence f₁, f_m be X when y = ∞ and X′ when y = -∞ . Then the number of roots of f(z) having their real parts positive, less the number having their real parts negative, is X′-X . If now there are v real roots of the common divisor f_m(y) of f₁ and f₂, then f(z) has (X′-X + n-r)/2 roots with the real part positive, (X-X′ + n-r)/2 with the real part negative, and r purely imaginary roots.