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The solution is a straightforward application of Integrate. Integrate[Exp[-((x - x0)^2 + (y - y0)^2)/(2 c) - I (kx x + ky y)], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, Assumptions -> c > 0] (* 2 c E^(-(1/2) c (kx^2 + ky^2) - I (kx x0 + ky y0)) π *)



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