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The T=0 dynamics of the one-dimensional s=1/2 planar antiferromagnet is studied by an approach which consists of exact analytic calculations in the Bethe formalism and numerical finite-chain calculations on rings up to 10 spins. The method makes use of well known critical exponents for the correlation functions and of exact sum rules. The authors obtain approximate analytic expressions for both the out-of-plane and the in-plane dynamic structure factors, and for related quantities such as integrated intensities, susceptibilities and autocorrelation functions. The results are discussed in relation to possible experiments on quasi-1D magnetic compounds at low T. The calculations make clear that the T=0 dynamic structure factors are dominated by two-parameter continua of excitations rather than by single branches of spin-waves as predicted by classical spin-wave theory. By varying the planar anisotropy the autocorrelation functions display interesting features in their longtime asymptotic behaviour, such as a crossover from a uniform power-law decay to an oscillatory decay and a crossover between oscillatory decays with different frequencies. They conjecture a possibility of approaching the classical limit s= infinity starting from the quantum limit s=1/2. This provides a qualitative, and in some aspects even quantitative, understanding of the dynamical behaviour of s>1/2 systems in terms of a quantum approach.