Running it on Mathematica 10
In[1]:= f[t_] := A (E^(-a t) - E^(-b t))
In[2]:= FourierTransform[f[t], t, \[Omega]]
Out[2]= A Sqrt[2 \[Pi]] DiracDelta[I a + \[Omega]] -
A Sqrt[2 \[Pi]] DiracDelta[I b + \[Omega]]
complex delta function DiracDelta[I a + [Omega]] means that function = 0 except at point [Omega]=-I a
