A textbook on electron optics states that, ignoring a factor of 2 for convenience, the result
$\mathscr{F}(I(\vec{r}))=\mathscr{F}(\phi(\vec{r}))\cdot{}A(\vec{k})\cdot{}\sin[\gamma(\vec{k})]$
can be derived from
$I(\vec{r})=\left|\mathscr{F}^{-1}\left(\mathscr{F}\left[\psi_0e^{i\phi(\vec{r})}\right]A(\vec{k})e^{i\gamma(\vec{k})}\right)\right|^2$,
where $i=\sqrt{-1}$ and $\mathscr{F}$ denotes Fourier transformation. The assumption $\phi(\vec{r})\ll1$ is applicable, so terms above the first order in the expansion of the exponential function can be ignored (i.e., $\exp[i\phi(\vec{r})]=1+i\phi(\vec{r})$). Could someone please show the details of this derivation? In particular, I am interested to see how $\exp[i\gamma(\vec{k})]=\cos[\gamma(\vec{k})]+i\sin[\gamma(\vec{k})]$ is reduced to $\sin[\gamma(\vec{k})]$. My justification for posting this on math.se rather than physics.se stems from the fact that the question relates specifically to the derivation of one equation from another and is therefore independent of the underlying physical principles. Nevertheless, if information about the physical interpretation of the symbols in the above relations is needed, a more detailed description of the problem is included below.
Possibly extraneous additional information:
The intensity distribution in the image plane is given by $I(\vec{r})=\lvert\psi_i(\vec{r})\rvert^2$, where $\psi_i(\vec{r})$, the wave function in the image plane, is given by, $\psi_i(\vec{r})=\mathscr{F}^{-1}\left(\mathscr{F}[\psi(\vec{r})]\cdot{}A(\vec{k})\cdot{}\exp[i\gamma(\vec{k})]\right)$, $\vec{r}$ is a real-space position vector, $\vec{k}$ is a Fourier-space spatial frequency vector, $\psi(\vec{r})=\psi_0\cdot{}\exp[i\phi(\vec{r})]$ is the wave function just behind the object to be imaged that has been phase-shifted by the amount $\phi(\vec{r})$ compared to the incident plane wave $\psi_0=\exp[ikz]$, $z$ is the direction of propagation of the incident plane wave, $k=|\vec{k}|$, and $A(\vec{k})$ and $\gamma(\vec{k})$ are functions in reciprocal space describing the aperture and wave aberrations associated with the objective lens, respectively.
The author states without proof that for $F(\vec{k})=\mathscr{F}(I(\vec{r}))$ (the Fourier transform of the image plane intensity) and $O(\vec{k})=\mathscr{F}(\phi(\vec{r}))$ (the Fourier transform of the phase shift, $\phi(\vec{r})$, which itself equals the projected three-dimensional Coulomb potential of the object being imaged), one finds that $F(\vec{k})=O(\vec{k})\cdot{}A(\vec{k})\cdot{}\sin[\gamma(\vec{k})]$.