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Fourier Transform Inversion Using an Elementary Differential Equation and a Contour Integral
Let f be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that f is absolutely continuous such that f and f′ are Lebesgue integrable. A function g is defined by f′(t)−iwf(t)=g(t). This differential equation has a well-known integral solution using the Heaviside …