Finding the transfinite diameter of the level sets of complex logarithm Given a simply-connected domain $|g(z)|\ge C$ how can I find the analytic conformal mapping guaranteed by the Riemann mapping theorem?
In particular I'm interested in finding the transfinite diameter of the domain $|\log z|\le C$ as a function of $C$. As I understand it—I had to look up the terms—this means finding a function $f$ mapping the closure of the exterior $|\log z|\ge C$ to the unit disk, with $f(\infty)=0$, and the the answer I want will be $f'(\infty).$
So what conformal mapping $f$ sends $|\log z|\ge C$ to the unit disk, or (if it's easier) what is $f'(\infty)$?
Edit: Hagen correctly pointed out that there are issues with $\log z$ near $\pi$. For my application you can assume that $0<C<3.$ In fact probably $C<1.$
Edit: I don't have a scan of the section of the paper this comes from, but here's the rough translation I wrote into my notes:

If $f(x)$ is an entire function, $S$ is the circle $|S|\le\gamma,$ the transform $T$ is the domain $|\log z|\le\gamma,$ and the transfinite diameter $\tau$ grows with $\gamma$ and is equal to 1 for $\gamma=\gamma_0=0.843\ldots,$ then [the desired result obtains.]

I did not write out the proof (or a translation) but it said very little, essentially just that $\gamma_0$ was what you got when you set the transfinite diameter to 1. It did not give the mapping or give any further information.
 A: It is amusing to observe that all later authors citing Pisot's result reproduce $0.843\dots$ with the same number of digits (Google Search for Pisot 0.843 turned up a few). None of them commented on how the number came about. A clue to Pisot's method may be found in the paper Über ganzwertige ganze Funktionen (1942) where he experiments with polynomials $p(z)=(z-1)^a(z-2)^b(z^2-3z+3)^c$ arriving at $a=6,  b=2, c=1$ presumably by trial and error. These are polynomials with roots around $1$, and the idea is that the level sets of $p$ approximate the level sets of logarithm.  This particular polynomial does not give precise bounds for transfinite diameter of level set. I still don't know how Pisot found $0.843\dots$ so precisely without computers; it's likely that his method was more sophisticated than what I present below.  
A key fact about transfinite diameter is that for any monic polynomial $p$ of degree $n$, the transfinite diameter of the set $E_\lambda=\{z:|p(z)|\le \lambda\}$ is equal to $\lambda^{1/n}$. The proof can be found, for example, in Potential theory in the complex plane by Ransford. For the special case when $E_\lambda$ is connected (which is the only one needed here) it suffices to observe that $(p(z)/\lambda)^{1/n}$ has a single-valued branch in the complement of $E_\lambda$, and maps the complement onto the exterior of unit disk. At infinity this function is asymptotic to $z/\lambda^{1/n}$, hence the result. 
Also, the transfinite diameter is monotone with respect to inclusion. Therefore, if $E$ is any set and $p$ is a monic polynomial such that $E_\lambda \subset E\subset E_\mu$, then we know the transfinite diameter of $E$ is between $\lambda^{1/n}$ and $\mu^{1/n}$. 
So, to estimate the transfinite diameter of the set $\{e^z:|z|\le r\}$, we must come up with polynomials that have all roots within this set and have nearly constant modulus on its boundary. I take $r=0.843$ for illustration. The set is shown below.

Pretty round, but with noticeable flatness on the left, and somewhat stretched out vertically. The flatness tells me to put some vertically spaces zeros on the left. On the right I might want to have a real zero. (Or not...) By trial and error, I found a fifth degree polynomial with decent fit: blue curve is the set $|p|=1$, the red curve is as above. 

The roots, marked above, $0.985\pm 0.46i$, $ 1.51\pm 0.51i$, and $ 1.79$. The blue curve is $|p|=1$ (implicitplot in Maple). 
To quantify the quality of fit,  look at the values $\{|p(e^z)|^{1/5}: |z|=0.843\}$. Computer tells me they are between $0.992$ and $1.005$. Hence, the transfinite diameter lies between these numbers as well. 
One possible way to improve precision is to use multiple roots, like Pisot did. 

As an aside, Thomas Ransford, the author of aforementioned book, is  an expert in high-precision computation of transfinite diameter (aka logarithmic capacity).  
