Fourier representation of aperiodic irregular function

[MadMax]

We have

$$\epsilon(i f, r) = \epsilon_2(i f)$$ when $$H + h_2(x) \leq z < + \infty$$
$$\epsilon(i f, r) = 0$$ when $$h_1(x) < z < H + h_2(x)$$
$$\epsilon(i f, r) = \epsilon_1(i f)$$ when $$- \infty < z \leq h_1(x)$$

show the corresponding Fourier transform is

$$\frac{i}{q_z} \int d^2x e^{iq_\bot \cdot x}[\epsilon_2 e^{iq_z[H+h_2(x)]} - \epsilon_1 e^{iq_z h_1(x)}]$$

I've looked in a few books but tbh I have no real idea how to show this...

Any help/suggestions/tips would be much appreciated. Thanks.

 

[MadMax]

a small hint please?