File:AnalyticTetrationBaseSqrt2u00.png: Difference between revisions

${\displaystyle ~G(z)}$

${\displaystyle b={\sqrt {2}}}$

on the complex ${\displaystyle ~z}$ plane,
Solution of equation ${\displaystyle \exp _{\sqrt {2}}(G(z))=G(z+1)}$

Levels of integer real part and those of integer imaginary part are shown with thick lines. Intermediate levels are shown with thin lines.

Grid covers with unit step range (${\displaystyle -10\leq \Re (z)\leq 10}$, ${\displaystyle -10\leq \Im (z)\leq 10}$).

Function is periodic; period ${\displaystyle T=17.1431~{\rm {i}}}$.

Asymptotic increment ${\displaystyle Q_{1}=\ln ^{2}(2)\approx -0.36651292058166432701}$ at ${\displaystyle \Re (z)\rightarrow \infty }$, and ${\displaystyle Q_{2}=\ln ^{2}(4)\approx 0.32663425997828098240}$ at ${\displaystyle \Re (z)\rightarrow -\infty }$.

Period: ${\displaystyle T={\frac {2\pi {\rm {i}}}{Q_{2}}}\approx 19.23614904204285471\!~i}$.

Asymptotic behavior:

${\displaystyle G(z)\sim 4+{\rm {e}}^{Q_{2}z}+o({\rm {e}}^{Q_{2}z})}$

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