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The explicit evaluation of the rate-distortion function for continnous-amplitude sources when it is strictly greater than its Shannon lower bound has been previously considered a formidable task. This is the case for independent identically distributed (i.i.d.) Gaussian sources under an absolute-magnitude distortion criterion. In this paper, we provide a method that yields the explicit evaluation of for this case. This method is then shown to be applicable to the evaluation of the absolute-magnitude-criterion for a large class of i.i.d, sources having probability densities with constrained tail decay. Other applications are discussed.