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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 Â¿ plane is used in the Taylor-Cauchy transform, a complex Â¿ plane is used in the Laurent-Cauchy trans-form where the two complex variables are related by Â¿= 1/Â¿. In the Taylor-Cauchy transform method, W(k)(Â¿), the kth derivative of W(Â¿), converges inside a circle. However, in the Laurent-Cauchy transform method, H(Â¿) converges outside of a circle of radius R. Thus, W(k)(Â¿) is expressible by a Taylor series whose general term is Â¿ wn,kÂ¿n and H(Â¿) is expressible by a Laurent series whose general term is Â¿ hn Â¿ -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.