# FourierParameters default differs from documentation?

Running Mathematica 11.2.0 I see in the documentation that FourierParameters defaults to {0,1}, yet when I try a Fourier transform with the default options, vs those values explicitly, I get different results. It looks like the defaults actually appear to be {0,-2Pi} :

FourierTransform[Sinc[t], t, \[Omega]]
FourierTransform[Sinc[t], t, \[Omega], FourierParameters -> {0, 1}]
FourierTransform[Sinc[t], t, \[Omega], FourierParameters -> {0, -2 Pi}]


These give respectively

$$\frac{1}{2} \pi \text{sgn}(2 \pi \omega +1)+\frac{1}{2} \pi \text{sgn}(1-2 \pi \omega )$$

$$\frac{1}{2} \sqrt{\frac{\pi }{2}} (\text{sgn}(1-\omega )+\text{sgn}(\omega +1))$$

$$\frac{1}{2} \pi \text{sgn}(2 \pi \omega +1)+\frac{1}{2} \pi \text{sgn}(1-2 \pi \omega )$$

Is FourierParameters configurable somehow, and is Mathematica finding a default FourierParameters value that differs from the documentation?

You can check the default options of any symbol with Options. For me, Options[FourierTransform] outputs:

{Assumptions :> $Assumptions, GenerateConditions -> False, FourierParameters -> {0, 1}} Which coincides with the documentation. Furthermore, the output I get without parameters matches the results for FourierParameters -> {0, 1}. If you Unprotect[FourierTransform], you can change the default options simply by setting them, e.g.: Unprotect[FourierTransform]; Options[FourierTransform] = {Assumptions :>$Assumptions, GenerateConditions -> False, FourierParameters -> {0, -2 Pi}};


However, this is difficult to actually recommend.

Something is weird with your setup:

FourierTransform[Sinc[t], t, w]
FourierTransform[Sinc[t], t, w, FourierParameters -> {0, 1}]
FourierTransform[Sinc[t], t, w, FourierParameters -> {0, -2 Pi}]

1/2 Sqrt[\[Pi]/2] (Sign[1 - w] + Sign[1 + w])
1/2 Sqrt[\[Pi]/2] (Sign[1 - w] + Sign[1 + w])
1/2 \[Pi] Sign[1 - 2 \[Pi] w] + 1/2 \[Pi] Sign[1 + 2 \[Pi] w]


Maybe you have some lingering definitions that conflict with the transform calculation.

• It seems that a one time execution of ClearAll[ [Omega] ] resolved this issue, as I did not see it when w was used. Curiously, before that ClearAll, even after restarting Mathematica I still saw the difference from the default with {0,1}. Commented Feb 13, 2018 at 13:58