(related to Proposition: Polar Coordinates of a Complex Number)
- By hypothesis, $z\in\mathbb C$ can is a complex number.
- If $z=0,$ we can write $z=0\cdot\exp(i\phi)$ with an arbitrary $\phi\in\mathbb R,$ with the complex exponential function $\exp:\mathbb C\to\mathbb C.$
- Therefore, let $z\neq 0.$
- Since $z\neq 0$, it has a positive absolute value $r:=|z| > 0.$
- Set $\zeta:=z/r,$ then $|\zeta|=1.$
- We can write this complex number as $\zeta=\xi+i\eta.$
- Note that $\xi^2+\eta^2=1.$
- By setting $cos(\phi):=\xi$ and $\sin(\phi):=\eta$, we can deduce from Euler's formula $$\zeta=\exp(i\phi)=\cos(\phi)+i\sin(\phi).$$
- It follows that $$z=r\exp(i\phi).$$
- It remains to be shown that this polar representation of the complex number $z$ is unique.
- But this follows from the general case of the Euler's identity by which from the equality of $\exp(i\phi)=\exp(i\psi)$ it follows that $\exp(i(\psi-\psi)=1.$
- It follows, $\exp(i\phi)$ is unique modulo $k\cdot 2\pi$ for all integers $k\in\mathbb Z.$
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983