Synthesis

\normaldensity (\varphi)= \sum\limits_{s = - \infty}^{\infty} \psi_s \frac{\exp(-\I s\varphi)}{2 \pi}

For the synthesis:

(1)   \begin{align*} \normaldensity (\varphi) &= \frac{\psi_0}{2\pi} + \frac{1}{2\pi}\sum\limits_{s = 1}^{\infty} [\psi_s \exp(-\I s\varphi)+\psi_{-s}\exp(\I s\varphi))] \\ &= \frac{\psi_0}{2\pi} + \frac{1}{2\pi}\sum\limits_{s = 1}^{\infty} [\psi_s \exp(-\I s\varphi)+\psi_{s}^*\exp(\I s\varphi))] \\ &= \frac{\psi_0}{2\pi} + \frac{\psi_0}{2\pi}\sum\limits_{s = 1}^{\infty} [q_s \exp(\I \arg\psi_s) \exp(-\I s\varphi)+q_s \exp(-\I \arg\psi_s)\exp(\I s\varphi))] \\ &= \frac{\psi_0}{2\pi} + \frac{\psi_0}{2\pi}\sum\limits_{s = 1}^{\infty} q_s [\exp(-\I (s\varphi-\arg\psi_s))+\exp(\I (s\varphi -\arg\psi_s))] \\ &= \frac{\psi_0}{2\pi} + \frac{\psi_0}{\pi}\sum\limits_{s = 1}^{\infty} q_s \cos(s\varphi-\arg\psi_s) \\ &= \frac{\psi_0}{2\pi}\left[1 + \sum\limits_{s = 1}^{\infty} 2 q_s \cos(s\varphi-\arg\psi_s)\right] \end{align*}