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When I can calculate an inverse fourier transform by the built-in function InverserFourierTransform, the result is different from what I calculate by definition through integration manually. and I noticed that if I give a negative sign in the Exp function, then the solution is exact the same.
See here:
The functions Fourier, FourierTransform, InverseFourier, and InverseFourierTransform (possibly others that I am not aware of) all accept a parameter called FourierParameters to determine the flavor of transform performed. By default, the value of FourierParameters is {0, 1}, which gives the results you've observed. To run a 'traditional' Fourier transform, you will need to set these to {0, -1}.
The explanation for these parameters can be found under the Details and Options section of this help page for the Fourier function (currently the 5th bullet). Basically, there are many conventions for how to perform the Fourier transform that are used in different situations and have similar properties to the convention that most of us learn in college; more information about these conventions can also be found on the wikipedia page for Fourier transforms.
One mildly confusing point: Fourier[x, FourierParameters -> {0,-1}] should give the same result as InverseFourier[x, FourierParameters -> {0,1}] due to the way the parameters are interpreted. Make sure to keep in mind that if you use FourierParameters -> {0, -1} with the forward function then you have to use it for the inverse as well; i.e.:
x == InverseFourier[
Fourier[x, FourierParameters -> {0,-1}],
FourierParameters -> {0, -1}];
(* => True *)
There appears to be a problem with version 10.1 compared to version 10.0
Your direct integration implies that your Fourier parameters are {-1, 1}
$Version
"10.0 for Mac OS X x86 (64-bit) (December 4, 2014)"
F[w_] = 1/(1 - I*w);
f[t_] = InverseFourierTransform[F[w], w, t,
FourierParameters -> {-1, 1}]
(2*Pi*HeavisideTheta[t])/E^t
F[w] == FourierTransform[f[t], t, w,
FourierParameters -> {-1, 1}] // Simplify
True
test = Simplify[f[t], #] & /@ {t < 0, t > 0}
{0, (2*Pi)/E^t}
Version 10.0 handles the direct integration easily
f2[t_] = Integrate[F[w]*Exp[-I*w*t],
{w, -Infinity, Infinity}, Assumptions -> Element[t, Reals]]
(Pi*(1 + Sign[t]))/E^Abs[t]
This is equivalent except for t==0 for which the HeavisideTheta is undefined
test == (Simplify[f2[t], #] & /@ {t < 0, t > 0})
True
However, with version 10.1 while the built-in functions work the direct integration does not
$Version
"10.1.0 for Mac OS X x86 (64-bit) (March 24, 2015)"
F[w_] = 1/(1 - I*w);
f[t_] = InverseFourierTransform[F[w], w, t, FourierParameters -> {-1, 1}]
(2*Pi*HeavisideTheta[t])/E^t
Simplify[f[t], #] & /@ {t < 0, t > 0}
{0, (2*Pi)/E^t}
F[w] == FourierTransform[f[t], t, w,
FourierParameters -> {-1, 1}] // Simplify
True
Version 10.1 does not handle the direct integration
f2[t_] = Integrate[F[w]*Exp[-I*w*t],
{w, -Infinity, Infinity}, Assumptions -> Element[t, Reals]]
(1/2)I((1/Sqrt[Pi])* (-MeijerG[{{1/2, 1, 1}, {}}, {{1}, {}}, (2*I)/Abs[t], 1/2] - I*MeijerG[ {{1/2, 1/2, 1}, {}}, {{1/2}, {}}, (2*I)/Abs[t], 1/2]*Sign[t]) + 2*CosIntegral[IAbs[t]] (-Cosh[Abs[t]] + Sign[t]* Sinh[Abs[t]]) - I*(Cosh[Abs[t]]Sign[t] - Sinh[Abs[t]])(Pi - 2*I*SinhIntegral[Abs[t]]))
FourierParametersfor consistency with your convention. $\endgroup$ – J. M.'s ennui♦ May 6 '15 at 9:07