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Evaluation of the Fourier integral by parts leads to two different series representations of a time function in the frequency domain. These series involve powers of the frequency variable and either derivatives or integrals of the function, evaluated at the upper time limit only. It is necessary that the function so treated be analytic between the time limits imposed. The convenience of the formulation then follows from choosing the time limits symmetrically. The method is presented as a mathematical experiment rather than a rigorous formulation.