imported>Dmitrii Kouznetsov |
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| == Summary == | | == Summary == |
| {{Image_notes_ownwork2
| | Importing file |
| |Description =Parameters of the asymptotic of tetration at base <math>b</math>
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| Absicssa for all curves is <math>\ln(b)</math>.
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| The scale in the ordinate axis is 0.1 of scale at the absciss axis.
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| Values
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| <math>\ln(b)\!=\!\ln(2)/2</math>,
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| <math>\ln(b)\!=\!1/\rm e</math>,
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| <math>\ln(b)\!=\!\ln(10)\!\approx\! 2.3</math>~
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| are shown with additional vertical gridlines;
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| ordinate equal to <math>\rm e</math> is shown with additional horisontal gridline.
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| \vskip 2mm
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| {\bf Eigenvalues of logarithm. }
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| Solutions of Equation
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| <math>L=\log_b(L)</math>
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| are plotted with thin lines;
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| At <math>\ln(b)<1/\rm e</math>, there exist two real solutions; the curve goes through points
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| <math>(\ln(2)/2,2)</math>,
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| <math>(1/\rm e,\rm e)</math>,
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| <math>(\ln(2)/2,4)</math> and passes close to point
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| <math>(0.3,6)</math>.
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| At <math>\ln(b)=1/e</math>, the two solutions coincide.
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| At <math>\ln(b)>1/e</math>, the two solutions differ only by the sign of the imaginary part;
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| the two options for the imaginary parts are shown with with thin dashed lines; and the
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| thin solid line indicates the real part.
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| At <math>\ln(b)>1.6</math>, the real part of
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| <math>L</math> is negative.
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| '''Asymptotic increment.'''
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| The thick lines shows the asymptotic increment <math>Q</math>.
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| At <math>\ln(b)<1/e</math>, the two possible values of increment are real; negative values indicate
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| that the asymptotic decays in the direction of real axis.
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| At <math>\ln(b)=1/e</math>, the increment is zero.
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| At <math>\ln(b)>1/e</math>, there are two possible values of increment, that differ by signum of its
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| imaginary part. The positive imaginary part is plotted with dashed line.
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| That with negative imaginary part is not plotted.
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| The real part of increment
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| <math>Q</math> is shown with thick solid line. At
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| <math>\ln(a)>1/\rm e</math>,
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| the real part is positive, and in the range of the figure it does not esceed unity.
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| '''Asymptotic period'''
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| The asymptotic period <math>T=2\pi i/Q</math> is shown with dotted lines.
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| At <math>\ln(b)<1/e</math> there are two possibel periods, and they have pure imaginary valies.
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| In order to simplify the comparison, the modulus of the period is plotted for the case of
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| negative decrement (lower branch of the thick curve).
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| At <math>\ln(b)=1/e</math>, both asymptotic periods are infinite.
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| At <math>\ln(a)>1/e</math>, there exist two mutually-conjugated solutions; the real part of the
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| period <math>T</math> is shown with the upper dotted curve, while the imaginary part is shown
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| with lower dotter curve. Again, the option with negative imahhinary part is not plotted
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| in order to avoid overfill the figure with curves.
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| |Year_created =2008
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| |CZ_username = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
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| |Notes =
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| |Other_versions =
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| }}
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| {{attribution}}
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| == Licensing/Copyright status ==
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| {{CC-by-sa-3.0}}
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