A Closed Expression for the Root Locus Method

In 1961 the author published a paper [2] concerning the paths of the zeros of f(z, K) = g(z) Kei0h(z) where g(z) = z' + aZn-1 +? h(z) = zm + bzm'+ * * * as K varies from 0 to oo under the assumptions that n > m, 0 is a real constant and g(z), h(z) are complex polynomials. Asymptotes were established as well as precise approximations to the zeros for K large and formulae for the direction of the paths of the zeros, i.e., root loci, as K approached 0 and oo. In addition various properties were discussed when f(x, K) was assumed to be a real polynomial, that is, f(z, K) had real coefficients. This paper proposes to find a closed expression satisfied by the coordinates of the root loci. Throughout it will be convenient to consider K as varying from oo to + oo instead of from 0 to + oo and to include the zeros of h(z), (i.e., permit K to be infinite) as part of the root locus. f(z, K) will also be written as