tries: $$k_2 = \frac{e^{z/ \lambda}}{2 \sinh{\sigma/\lambda}} \sim  \frac{const}{\sigma}.$$ Above agrees qualitatively with result presented in Fig. \[Convergence3\] if $\sigma \ll z$.

However, statistical distribution of $\exp-\log$ numbers is unknown, PDF was chosen intuitively. Statistical properties of EL numbers should not depend on on language used to generate them, at least in the limiting case of large complexity. Numerical evidence, obtained by collecting real numbers generated according to procedure presented in Sect. \[enum\], is presented in Fig. \[StatReals\]. Numbers with $\operatorname{Im}(z) \neq 0$ were discarded[^4] to simplify analysis and presentation.

![\[StatReals\] Histogram of first 20k real numbers generated with use of calculators 1-4. Dashed line show exponentially transformed Cauchy distribution . ](figs/StatReal){width="50.00000%"}

None of the widespread distributions match numbers presented in Fig. \[StatReals\]. The most close are: Pareto, Levy and Cauchy distributions. Noteworthy, all of them have undefined mean and infinite dispersion. If we look at statistics of base 10 logarithm, it mimic t-Student, Cauchy and Laplace (double $\exp$) distributions. Probably this is indeed a brand new statistics, following some empirical distribution. True distribution of real EL numbers (Fig. \[StatReals\]) is, however, quite well described by transformed Cauchy distribution with PDF: $$\label{cauchy}
P(x) = \frac{1}{\pi x} \frac{1}{1+(\ln{x})^2}.$$ Therefore, later, statistical distribution 