Problem with a Fourier Transform result

Questions
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1

Consider the following code:

In[22]:= FourierTransform[Exp[I u t],t, \[Omega]]
    Out[22]= Sqrt[2 \[Pi]] DiracDelta[u+\[Omega]]

While,

In[25]:= FourierTransform[Sqrt[2 Pi]DiracDelta[u+\[Omega]],\[Omega],t]
Out[25]= E^(-I t u)

In addition the coefficient are supposed to be 2 Pi instead of Sqrt[2 Pi]

2

Have you looked at option: FourierParameters, i’m quite sure that is the ‘issue’:

FourierTransform[Sqrt[2 Pi] DiracDelta[u + \[Omega]], \[Omega], t, FourierParameters -> {1, -1}]
3

I think you should use the InverseFourierTransform

In[11]:= f = FourierTransform[Exp[I u t], t, \[Omega]]
InverseFourierTransform[f, \[Omega], t]

Out[11]= Sqrt[2 \[Pi]] DiracDelta[u + \[Omega]]

Out[12]= E^(I t u)