Algorithms are described for the accurate evaluation of a class of integrals, including Fourier transforms, in which the integral is well represented as the product of an exponential and a polynomial. The algorithms treat all or a part of the exponential dependence exactly and represent the polynomial by conventional finite difference. Numerical examples of Fourier transforms are given that illustrate the accuracy and stability of the algorithms and include a case in which the evaluation of Fourier transforms does not require the evaluation of any sine, cosine, or complex exponential functions.

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