Laurent-Cauchy Transforms for Analysis of Linear Systems Described by Differential-Difference and Sum Equations

The Taylor-Cauchy transform for the analysis of nonlinear systems was presented at the 1959 IRE National Convention. In this paper it is shown that the Laurent-Cauchy transform can be derived from the Taylor-Cauchy transform by a simple mapping. Whereas a complex λ plane is used in the Taylor-Cauchy transform, a complex ρ plane is used in the Laurent-Cauchy trans-form where the two complex variables are related by λ = 1/ρ. In the Taylor-Cauchy transform method, W(k)(λ), the kth derivative of W(λ), converges inside a circle. However, in the Laurent-Cauchy transform method, H(ρ) converges outside of a circle of radius R. Thus, W(k)(λ) is expressible by a Taylor series whose general term is Σ wn,kρn and H(ρ) is expressible by a Laurent series whose general term is Σ hn ρ-n. In either case, the relation between the coefficients of the power series and the function of a complex variable is given by Cauchy's integral. In conjunction with the Laplace transform, the Laurent-Cauchy transform can be used to analyze discrete-continuous systems such as digital servomechanisms, retarded feedback control systems, certain analog-digital computer systems, and pulsed-data systems. If n is replaced by n T, where T is a sampling time, and hn is taken to mean h(n T), the Laurent-Cauchy transform reduces to the Z-transform. A table of Laurent-Cauchy transforms, several theorems, and three examples are given in the paper.