File:TetrationAsymptoticParameters01.jpg

${\displaystyle b}$

Absicssa for all curves is ${\displaystyle \ln(b)}$. The scale in the ordinate axis is 0.1 of scale at the absciss axis. Values ${\displaystyle \ln(b)\!=\!\ln(2)/2}$, ${\displaystyle \ln(b)\!=\!1/{\rm {e}}}$, ${\displaystyle \ln(b)\!=\!\ln(10)\!\approx \!2.3}$~ are shown with additional vertical gridlines; ordinate equal to ${\displaystyle {\rm {e}}}$ is shown with additional horisontal gridline. \vskip 2mm

Eigenvalues of logarithm. Solutions of Equation ${\displaystyle L=\log _{b}(L)}$ are plotted with thin lines; At ${\displaystyle \ln(b)<1/{\rm {e}}}$, there exist two real solutions; the curve goes through points ${\displaystyle (\ln(2)/2,2)}$, ${\displaystyle (1/{\rm {e,{\rm {e)}}}}}$, ${\displaystyle (\ln(2)/2,4)}$ and passes close to point ${\displaystyle (0.3,6)}$. At ${\displaystyle \ln(b)=1/{\rm {e}}}$, the two solutions coincide. At ${\displaystyle \ln(b)>1/{\rm {e}}}$, the two solutions differ only by the sign of the imaginary part; the two options for the imaginary parts are shown with with thin dashed lines; and the thin solid line indicates the real part. At ${\displaystyle \ln(b)>1.6}$, the real part of ${\displaystyle L}$ is negative.

Asymptotic increment. The thick lines shows the asymptotic increment ${\displaystyle Q}$. At ${\displaystyle \ln(b)<1/e}$, the two possible values of increment are real; negative values indicate that the asymptotic decays in the direction of real axis. At ${\displaystyle \ln(b)=1/e}$, the increment is zero. At ${\displaystyle \ln(b)>1/e}$, there are two possible values of increment, that differ by signum of its imaginary part. The positive imaginary part is plotted with dashed line. That with negative imaginary part is not plotted. The real part of increment ${\displaystyle Q}$ is shown with thick solid line. At ${\displaystyle \ln(a)>1/{\rm {e}}}$, the real part is positive, and in the range of the figure it does not esceed unity.

Asymptotic period The asymptotic period ${\displaystyle T=2\pi i/Q}$ is shown with dotted lines. At ${\displaystyle \ln(b)<1/e}$ there are two possibel periods, and they have pure imaginary valies. In order to simplify the comparison, the modulus of the period is plotted for the case of negative decrement (lower branch of the thick curve).

At ${\displaystyle \ln(b)=1/{\rm {e}}}$, both asymptotic periods are infinite.

At ${\displaystyle \ln(a)>1/{\rm {e}}}$, there exist two mutually-conjugated solutions; the real part of the period ${\displaystyle T}$ is shown with the upper dotted curve, while the imaginary part is shown with lower dotter curve. Again, the option with negative imahhinary part is not plotted in order to avoid overfill the figure with curves.

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