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The least-mean-square (LMS) algorithm is a useful and popular procedure for adaptive signal processing of both real-valued and complex-valued signals. Past analysis of the complex LMS algorithm has assumed that the input signal vector is circularly-distributed, such that the pseudo-covariance matrix of the input signal is zero. In this paper, we relax this assumption, providing a complete mean and mean-square analysis of the complex LMS algorithm for non-circular Gaussian signals. Our analysis unifies the statistical descriptions of the conventional (real-valued) LMS and complex LMS algorithms as specific cases of our more-general behavioral description, negating the need for a distinction between these two procedures. Simulations indicate that our analysis more-accurately predicts the behavior of complex LMS for non-circular signals as compared to existing analyses in the scientific literature.