Is it possible to decompose any complex number $z = x + iy\in \mathbb{C}$ with $0\leq|z|\leq2$ into a sum of two unitary modulus exponentials ? i.e. $ z = e^{i\phi_1} + e^{i\phi_2}$ ?
I tried to decompose the problem $x + iy = \cos(\phi_1) + \cos(\phi_2) + i(\sin(\phi_1) + \sin(\phi_2)) $ into a set of two real equations but is seems that they are not linear :
\begin{eqnarray}
\cos(\phi_1) + \cos(\phi_2) & = &x \\
\sin(\phi_1) + \sin(\phi_2) & = & y
\end{eqnarray}
If it is possible, are there any known algorithm ? I tried the usual trigonometric transformations without success. And formulating the problem in terms of modulus and phase rather than real and imaginary parts made it seem more complex.
Thanks in advance.
Best Answer
This document contains a closed-form solution for a slightly more general version of this problem: find $\theta_1,\theta_2$ such that $$\alpha_1\exp(i\theta_1) + \alpha_2\exp(i(\theta_2+\theta_1))=x+i y$$ with $\alpha_1,\alpha_2>0$ given, and $\alpha_1=\alpha_2=1$ in the context of this question.
The solution is $$\theta_2=\arccos\left(\frac{x^2+y^2-\alpha_1^2-\alpha_2^2}{2\alpha_1\alpha_2}\right)$$ $$\theta_1=\arctan(x/y) - \arctan\left(\frac{\alpha_2\sin(\theta_2)}{\alpha_1+\alpha_2\cos(\theta_2)}\right)$$