Complex output of Fourier transform from real-valued function Hi I'm struggling to understand what the output of a Fourier transform $\widetilde{f}(p)$ actually means.
The Fourier transform $\widetilde{f}(p)$ of a function $f$ where $f(x) = \sin(x)$ for $|x|<\pi$ and $0$ otherwise can be shown to be.
$$\widetilde{f}(p) =  \dfrac{2i\sin(p\pi)}{p^2-1}$$
Now my understanding of the Fourier transform was that it acted as a frequency analysis function. $\widetilde{f}(p)$ told us effectively how much of the frequency $p$ existed in the function but now I'm realising that doesn't quite make sense. For example there is a factor of $i$, so $\widetilde{f}(0.5) =8i/3$ but what does that mean? We were taught that the value of $\widetilde{f}(p)$ was analogous to the Fourier coefficients in the Fourier series of a periodic function but complex coefficients in the Fourier series of a real-valued function doesn't make sense.

The complex argument of a frequency component indicates the phase of the complex sinusoid in the spatial domain, referenced to $x = 0$.
Look at the Fourier Transform of $\cos(x - \phi)$ using the shift theorem
$$\mathscr{F}\left[\cos(x - \phi)\right] = \mathscr{F}\left[\cos(x)\right]e^{-i\phi p}$$ 
The shift of the cosinusoid manifests as a complex phase change for that consinusoid's component's magnitudes in the frequency domain. 
