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]]))
FourierParameters
for consistency with your convention. $\endgroup$ – J. M.'s ennui♦ May 6 '15 at 9:07