Mapping spiral to imaginary axis

Question

Does anybody know how to map the complex spiral: $$ e^{i \omega n + \sigma n} $$ onto an imaginary line with a certain real part? Alternatively, what other ways are there of representing complex spirals?

Answer 1

I'm surprised that no one has answered this question. Your equation is almost a logarithmic spiral with a flair coefficient of $\sigma$. The logarithmic spiral should have $\sigma n \to \sigma\omega n$.

In general, most spirals in the complex plane have a form

$$z=f(\theta)e^{i\theta}$$

The most common spirals are the logarithmic, say $f(\theta)=e^{b\theta}$, and the Archimedean family, $f(\theta)=\theta^n$. When $n=1$, we have Archimedes's spiral, when $n=1/2$, Fermat's spiral, and $n=-1$, the hyperbolic spiral, to name a few. In polar coordinates we would say the radius is $r=f(\theta)$; $e^{i\theta}$ is just a circle in the complex plane.