In[]:=
arg=(1/Sqrt[2a])E^((-(x-b)^2)/(4a))
Out[]=
-
2
(-b+x)
4a
2
a
Pure Mathematics definition of Fourier transform
In[]:=
Simplify[FourierTransform[arg,x,k,FourierParameters{1,1}],a>0]
Out[]=
k(b-ak)
2π
Alternate “physicist” definition of Fourier transform
In[]:=
Simplify[FourierTransform[arg,x,k,FourierParameters{0,1}],a>0]
Out[]=
k(b-ak)