This question is related to one I asked here about the logarithmic spiral.
In the linked problem, I had to find and sketch the image of the straight line $z=(1+ia)t+ib$, for $-\infty < t < +\infty$,- where $a,b\in \mathbb{R}$ and $a \neq 0$ under the map $w=e^{z}$.
Substituting that expression for $z$ into $w = e^{z}$, and then eliminating the parameter, $t$, by letting $\varphi = at+b$, we obtained that $w = e^{\frac{\varphi}{a}-\frac{\varphi}{b}}e^{i\varphi}$, and that the modulus of $w$ gives us $r = Ce^{\varphi/a}$, where $C = e^{-b/a}$.
Now, I am asked to do the following:
Given $a \in \mathbb{R}$, find the condition on real numbers $C$, $C^{\prime} \,>0$ for the spirals $r=Ce^{\varphi/a}$ and $r = C^{\prime}e^{\varphi/a}$ to be the same (as sets on a plane). I.e., find conditions on $C$ and $C^{\prime}$ such that the logarithmic spiral is self-similar.
In doing some research about the logarithmic spiral, I have discovered that scaling by a factor of $e^{2\pi b}$, with the center of scaling at the origin, will give us back our original curve, but I am not sure why or how to show this algebraically.
Could somebody please help me out with this?
Thank you.
The differential equation
$$ \frac{r d\varphi}{ dr } = \tan \alpha $$
gives a log spiral by integration.
$$ r = C e^ { \cot \alpha \, \varphi} $$
As $C$ is an arbitrary constant, all log spirals are self similar, are identical upto a scaling factor.
The same can be found by complex number multiplication of arbitrary radius vectors with rates of progressions $a,b$. The same holds for $a=b$
$$ z_1 z_2 = c_1 e^{i a\varphi}\cdot c_2 e^{i b\varphi} = C e^{i \cot \alpha \, \varphi }$$
The log spirals are self similar.