File:LogConvergenceAnim.gif

LogConvergenceAnim.gif(500 × 516 pixels, file size: 210 KB, MIME type: image/gif, looped, 25 frames, 3.4 s)

Summary

Description
English: A plot showing the convergence of the power series about 1 for log at ${\displaystyle e^{i(\pi -{\frac {1}{3}})}}$.
 This mathematical image was created with Mathematica.
Date
Source Own work
Author Kan8eDie

Source

 Export["anim.gif", 

Table[ ListPlot[ Table[ it = Normal[Series[Log[z + 1] , {z, 0, n}]] /. z -> Exp[(\[Pi] - 1/3) I]; {Re[it], Im[it]}, {n, 0, a} ], PlotRange -> {0, 1.8}, Joined -> True, Mesh -> All, AxesLabel -> {Style[\[GothicCapitalR], Medium], Style[\[GothicCapitalI], Medium]}, AspectRatio -> 1, ImageSize -> 500, Epilog -> {it = Log[1 + Exp[(\[Pi] - 1/3) I]]; Darker[Blue], Text[ Style[ HoldForm[ Subscript[z, n] = Sum[(-1)^(k + 1) \[ExponentialE]^(k i (\[Pi] - 1/3))/k, {k, 1, n}]], 15, Antialiasing -> True ], {-0.8, 0.5} ], Red, PointSize[0.01], Point[{Re[it], Im[it]}], Text[Style[ HoldForm[ Log[1 + \[ExponentialE]^(i (\[Pi] - 1/3))] = Sum[(-1)^(k + 1) \[ExponentialE]^(k i (\[Pi] - 1/3))/k, {k, 1, Infinity}]], 15 ], {Re[it], Im[it] - 0.2}, {-0.6, 1}] } ], {a, 5, 77, 3}], "DisplayDurations" -> PadLeft[{1}, 25, 0.1] ] 

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 I, the copyright holder of this work, release this work into the public domain. This applies worldwide.In some countries this may not be legally possible; if so:I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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current09:32, 10 February 2009500 × 516 (210 KB)Kan8eDie (talk | contribs){{Information |Description={{en|1=A plot showing the convergence of the power series about 1 for log at $e^{i (\pi - \frac{1}{3})}$. {{Mathematica}}}} |Source=Own work by uploader |Author=Kan8eDie |Date=2008-02-10 |Permission=
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