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There is an obvious mistake in your code. You write f[x_] instead of f[k_], because your variable is k, not x, according to how you write things. On the second place, you should modify your code and write f[k_]:= A Exp[- a Abs[k] - I b k];. If you do so, the Fourier transform becomes g[x_] := Assuming[a > 0 && A \[Element] Reals && b \[...


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ComplexExpand is powerful. Clear["`*"]; f[x_] := E^(-(a + I*b) x); f[x] // ComplexExpand Conjugate[f[x]] // ComplexExpand


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As you want a real FFT, the picture must by symmetric or anti-symmetric. Here is a simple symmetric example: dat = Table[If[x^2 + y^2 < 4, 0, 1], {x, -n, n}, {y, -n, n}]; MatrixPlot[dat] fdat = FourierDCT[dat] // Chop; MatrixPlot[fdat] Here a more complicated one: n = 5; dat = Table[If[OddQ[x + y], 0, 1], {x, -n, n}, {y, -n, n}]; MatrixPlot[dat] fdat = ...


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