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Two-dimensional window functions and their Fourier transforms are widely used in electromagnetics and optics. It is shown that the two-dimensional Fourier transform of a window function can be written as a line integral along the boundary contour of that window function. This reduction of the two-dimensional Fourier integral to a one-dimensional line integral is attractive for both the closed-form and numerical evaluation of the Fourier transform of complex shaped window functions. Some applications of this new evaluation method are discussed.