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Integrate using properties of Fourier Transforms?

Posted 5 years ago
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Problem: Integrate over $x$ the following integral

$\int e^{-i\, \text{t1} \omega _{\text{p1}}-i \text{p1} x-i \text{t2} \omega _{\text{p2}}-i \text{p2} x}$

Is the following approach correct?

FourierTransform[(-1/Sqrt[2 \[Pi]]) E^(-I p1 x - I p2 x - 
   I t1 Subscript[\[Omega], p1] - 
   I t2 Subscript[\[Omega], p2]), x, (-p1 - p2)]

which yields

-(1/2) E^(-I t1 Subscript[\[Omega], p1] - 
  I t2 Subscript[\[Omega], p2]) DiracDelta[p1 + p2]

or in LaTeX form

$-\frac{1}{2} \delta (\text{p1}+\text{p2}) e^{-i \text{t1} \omega _{\text{p1}}-i \text{t2} \omega _{\text{p2}}}$

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