How to find isothermal coordinates equivalent to circles in far limit?

Question

I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of large distances from the origin, and 2) for the $\phi$=constant plane ($\phi$ is the azimuthal coordinate with $0\leq \phi \lt 2\pi$), the remaining coordinates $(u,v)$ are isothermal. So in this plane the standard cylindrical coordinates are $(\rho,z)$, $z$ being the axis of rotation, and $\rho$ being the normal distance from that axis. Define the following complex variable $\psi$.$$\psi=z+i\rho=re^{i\theta}=r(cos\theta+isin\theta)$$with $0\leq \theta\leq\pi$ and $(r,\theta)$ being the standard spherical coordinates. In that far limit, for the $\phi$ = constant plane, one of the coordinates is a family of circles with origin as center, and the other coordinate's asymptotes are a family of straight lines through the origin. It seems that the easiest way to study this is with conformal transformations $\psi=F(w)$ where $w=u+iv$. Then $\rho$ and $z$ both satisfy the 2-D Laplacian (and so do $u$ and $v$ ). In texts I can only find four functions which work (all have symmetry about the $z=0$ plane): $F(w)=$\begin{cases}e^w\\\cosh(w)\\\sinh(w)\\\sqrt(e^w+1)\\\end{cases}which in the case of rotational coordinate systems correspond respectively to the following : spherical, oblate and prolate spheroidal, and Cassinian ovoidal. The first 3 functions are derivable as the first term of the solution of the Laplacian by separation of variables (easy), and the last one seems to be a special solution of the Laplacian. Is it significant that these systems all involve only the exponential function (cosh and sinh are sums of them)? I would think that there are an infinite number of such systems, but how does one find them? By studying the solutions to the 2-D Laplacian, as infinite series or integrals?, or by conformal analysis?, or homotopy? These things can get very messy very quickly, so something with insight is probably needed. As a physicist, this seems to me like a very useful question, but way beyond my abilities. Any graduate students out there who need a good thesis question? Or is there an easy way?

Answer 1

Let me give what I think is an infinite set of such systems, and you can tell me whether it's what you had in mind: $e^w + p(e^w)e^{-(k+1)w}$, where $p(w)$ is a polynomial of degree at most $k$.

The first three examples you listed correspond, up to constants, to $p(w) = 0$, $p(w) = 1$, and $p(w) = -1$, respectively.

For $u=Re(w)$ large, $|e^w|$ is large, so $|p(e^w)e^{-(k+1)w}|\leq C|e^{-w}|$ is small. Therefore $e^w + p(e^w)e^{-(k+1)w}$ is quite close to the spherical-coordinates transformation $e^w$, so the asymptotic behaviour is circles-and-lines.