# Shell-Thron region

Shell-Thron region (Область Тронной Ракушки) is subset of set of complex numbers $$b$$ such that there exist limit

$$\displaystyle L = \lim_{n\rightarrow \infty} \exp_b^n(1)$$

This $$L$$ is a fixed point of $$\exp_b$$.

The Shell-Thron region is shown in figure at right; it is area inside the loop.

## William Harold Paulsen

In 2018, William Harold Paulsen suggests the following definition :

We say that the base $$b$$ is in the Shell-Thron region if the sequence of values $$\{ b, b^b, b^{b^b}, b^{b^{b^b}}, ... \}$$ converge to a finite fixed point.

## Evaluation of tetration

For $$b \in$$ Shell-Thron region, tetration $$\mathrm{tet}_b$$ can be evaluated through its asymptotyc expansion at large values of the real part of its argument, using the transfer equation

$$\exp_b(\mathrm{tet}_b(z))=\mathrm{tet}_b(z\!+\!1)$$

For values $$b$$ outside this region, the asymptotic expansion at complex fixed points of exponential can be used for large negative values of argument of the tetration. Then, the appropriate requirements on asymptotic behavior of tetration are postulated in order to provide its uniqueness; the tetration should approach its fixed points at large values of the imaginary part of its argument.

For $$v$$ at the margin of the Shell-Thron region, none of the methods above is sufficient to evaluate $$\mathrm{tet}_b$$, although the representation through the Cauchi integral still works , except case of real $$b \le \exp(1/\mathrm e) \approx 1.44466786101$$ ; then the special expansion is required 

In particular, the representation of tetration through the Cauchi integral  can be used for the Sheldon base, specific value $$b=1.52598338517 + 0.0178411853321~ {\rm i}$$, that is close to the margin of the Shell-Thron region, see Tetration to Sheldon base . However, no any specific difficulties in use of the Cauchi integral for the evaluation is detected, as base $$b$$ approaches the margin of the Shell-Thron region