imported>Caesar Schinas |
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| == Summary == | | == Summary == |
| {{Image notes ownworkpro
| | Importing file |
| |Description=plot of deviation of holomorphic [[tetration]] from its asymptotic expansion at large values of the imaginary part of the argument. (Real part of the argument is assumed to have moderate values).
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| Tertation tet is solution of equations
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| <math>\mathrm{tet}(z\!+1\!)=\exp\big(\mathrm{tet}(z)\big)</math>
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| <math>\mathrm{tet}(0)=1</math>, holomorphyc in the complex plane except the cut at the part of the real axis.
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| At large values of the imaginary part of the argument, tetratiomn can expanded asumptoticaly,
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| <math>\mathrm{tet}(z)=\sum_{n=0}^N a_n \varepsilon^n + \mathcal(O)\!\big(\varepsilon^{N\!+\!1}\big)</math>
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| where <math>\varepsilon=\varepsilon(z)=\exp(L z +R) </math> ;
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| <math>L\approx 0.2+1.3\mathrm{i}</math> is [[fixed point]] of logarithm,
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| and <math>R\approx 1.077961437526 -0.946540963949 \mathrm{i}</math> is universal constant.
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| First coefficients are:
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| <math>a_0=L</math>,
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| <math>a_1=1</math>,
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| <math>a_2=\frac{1/2}{L-1}\approx -0.151314 - 0.2967488 \mathrm{i}~,~</math>
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| <math>a_3=\frac{a_2+1/6}{L^2-1}\approx -0.36976 + 0.98730 \mathrm{i}</math><br>,
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| The four pictures show the deviation of [[tetration]] from the approcimation at
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| <math>N\!=\!0</math>,
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| <math>N\!=\!1</math>,
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| <math>N\!=\!2</math>, and
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| <math>N\!=\!3</math>; id est,
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| <math>f=\mathrm{tet}(z)-\sum_{n=0}^N a_n \varepsilon^n</math>.
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| The distribution of <math>f</math> in the complex <math>z</math>-plane is shown with lines
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| of constant phase and lines of constant modulus of <math>f</math>.
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| :Levels <math>\arg(f)=-2,-1</math> are shown with thick red lines.
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| :Level <math>\arg(f)=0</math> is shown with thick black lines.
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| :Levels <math>\arg(f)=1,2</math> are shown with thick blue lines.
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| :Levels <math>\arg(f)=\pm \pi</math> are shown with scratched lines.
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| :Levels <math>|f|=\exp(-0.8),\exp(-0.6),\exp(-0.4),\exp(-0.2)</math> are shown with thin red lines.
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| :Levels <math>|f|=\exp(0.2),\exp(0.4),\exp(0.6),\exp(-0.8)</math> are shown with thin blue lines.
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| :Levels <math>|f|=\exp(3), \exp(2), \exp(1),\exp(0), \exp(-\!1), \exp(-\!2), \exp(-\!3), \exp(-\!4), \exp(-\!5), \exp(-\!5), \exp(-\!7), \exp(-\!8)</math> are shown with thin thick black lines.
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| :Level <math>|f|=\exp(-10)</math> is shown with thick red line.
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| :Levels <math>|f|= \exp(-12),\exp(-14), \exp(-16),\exp(-18)</math> are shown with thick black lines.
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| :Level <math>|f|=\exp(-20)</math> is shown with thick red line.
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| :Levels <math>|f|= \exp(-22),\exp(-24), \exp(-26),\exp(-28)</math> are shown with thick black lines.
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| :Level <math>|f|=\exp(-30)</math> is shown with thick green line.
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| :Level <math>|f|=\exp(-31)</math> is shown with thick black line.
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| The plotter tried to draw also
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| :Level <math>|f|=\exp(-32)</math> with thick black line and
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| :Level <math>|f|=\exp(-33)</math> with thin dark green line, but they happened a littel bit scratched.
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| In the upper left corner of the last two pictures; the plotter could not distinguish the tetration from its asymptotic approximation.
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| |CZ username= Dmitrii Kouznetsov
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| |Year created= 2008
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| |Notes=is supposed to be used in the article [[Tetration]]
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| |Other versions=}}
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| == Licensing/Copyright status ==
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| {{CC|by-nd|3.0}}
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