File:TetrationAsymptoticExpansion00.jpg: Difference between revisions

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