The Riemann surface associated with $\log z$ can be naturally thought of as having elements of the form $r \angle \theta$, where $r \in \Bbb R$ and $r > 0$, and $\theta \in \Bbb R$.
This Riemann surface is naturally equipped with a product, such that $(r_1 \angle \theta_1) \cdot (r_2 \angle \theta_2) = (r_1 r_2) \angle (\theta_1 + \theta_2)$. It is fairly interesting in that it is the natural space that results when looking at complex numbers with "unwrapped" phases, which is a fairly basic notion that arises pretty frequently (such as in Fourier analysis).
It is also naturally equipped, as is every Riemann surface, with a metric $d(\cdot, \cdot)$. I am looking for what this metric is. In particular, I have the following two questions:
- Given two elements $a_1 = (r_1 \angle \theta_1)$ and $a_2 = (r_2 \angle \theta_2)$, what is a closed-form expression for $d(a_1, a_2)$ in terms of $r_1, \theta_1, r_2,$ and $\theta_2$?
- Given two elements $a_1$ and $b_1$, what is a closed-form expression for the midpoint between them using the above metric - e.g. the point of minimum and equal distance to both?
I think there should be answers for both of the above due to the nice properties of Riemann surfaces. I am not sure what the curvature of the metric should be, however...
That Riemann surface is conformally equivalent to the complex plane, which one can see by identifying it with the domain of the complex exponential map $$\exp : \mathbb C \to \mathbb C - \{0\} $$ The formula for that map is $$\exp(x+iy) = e^x \cos(y)+ i \, e^x \sin(y) $$ and so the change of coordinates between $z=x+iy \in \mathbb C$ and $(r,\theta) \in (0,\infty) \times (-\infty,+\infty)$ is given by $$r = e^x, \quad y = \theta $$ "The" metric on that Riemann surface, i.e. "the" conformal metric of constant curvature which in this case is curvature $0$, is therefore not quite unique, but it is unique up to scaling, and it is the Euclidean metric in $(x,y)$ coordinates given by $$d((x_1,y_2),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2} $$ One can then convert this into $r,\theta$ coordinates using the inverse coordinate change map $$x = \log(r), \quad \theta = y $$ to get $$d((r_1,\theta_1),(r_2,\theta_2)) = \sqrt{(\log(r_1)-\log(r_2))^2 + (y_1-y_2)^2} $$ Given that the midpoint in $z=x+iy$ coordinates is $((x_1+x_2)/2,(y_1+y_2)/2)$, one can then convert that with ease into $(r,\theta)$ coordinates as well.