Fourier Transform of A*(e^(-a*t)-e^(-b*t))

Questions
1

What is the Fourier Transform of
f(t) = A*(e^(-at)-e^(-bt))?

Typing it into Wolfram Alpha you get:
sqrt(2 pi) C delta(i a+omega)-sqrt(2 pi) C delta(i b+omega)

Question 1) Is this correct?
Question 2) What is a delta function when argument is complex?

Now consider

What is stopping me from using this solution to solve the problem. I.e:

Ft(w) = (A/Pi) (-a/Pi/(w^2 + (a/Pi)^2) - -b/Pi/(w^2 + (b/Pi)^2))

Question 3) What is the significance of k0 with reference to k? Or is k0 just a constant?

2

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