Now the graph makes sense and the rules of the game are on our hands. To find the Fourier transform of that function, you have to consider two cases:
Case 1: ω=0
F{f(t)}=2π1∫−∞∞f(t) eiωt dt
Case 2: ω=0
F{f(t)}=2π1∫−∞∞f(t) dt
Unlike the Laplace transform, the calculations in the Fourier transform are longer and need more attention!
Note: Some authors use
2π1 with the inverse Fourier transform, other authors even use
eiωt for the inverse Fourier transform. It does not matter much which notation you will use as along as you will be consistent with the two transforms (Fourier transform and Inverse Fourier transform). For a precise Fourier transform notation, follow your book definition.