File:ExpIPi.gif

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ExpIPi.gif ‎(360 × 323 pixels, file size: 11 KB, MIME type: image/gif, looped, 9 frames, 4.5 s)

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Summary[edit]

Description This is a demonstration that Exp(i*Pi)=-1 (called Euler's formula, or Euler's identity). It uses the formula (1+z/N)^N --> Exp(z) (as N increases). The Nth power is displayed as a repeated multiplication in the complex plane. As N increases, you can see that the final result (the last point) approaches -1, the actual value of Exp(i*pi).
Date
Source Own work
 
This diagram was created with Mathematica.
Author Sbyrnes321

Licensing[edit]

Public domain 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. 
(* Source code written in Mathematica 6.0, by Steve Byrnes, 2008. I release this code into the public domain. *)

plot1 = Table[
  ListPlot[Table[{Re[(1 + (\[ImaginaryI] \[Pi])/n)^m], 
     Im[(1 + (\[ImaginaryI] \[Pi])/n)^m]}, {m, 0, n}], 
   PlotJoined -> True, PlotMarkers -> Automatic, 
   PlotRange -> {{-2.5, 1.1}, {0, \[Pi] + .05}}, AxesOrigin -> {0, 0},
    AxesLabel -> {"Real part", "Imaginary part"}, 
   PlotLabel -> "N = " <> ToString[n], 
   AspectRatio -> Automatic], {n, {1, 2, 3, 4, 5, 10, 20, 50, 100}}];

Export["ExpIPi.gif", plot1, "DisplayDurations" -> {2}, 
 "AnimationRepititions" -> Infinity ]

File history

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Date/TimeThumbnailDimensionsUserComment
current19:46, 25 March 2010Thumbnail for version as of 19:46, 25 March 2010360 × 323 (11 KB)Aiyizo (talk | contribs)optimized animation, converted to 16 color mode
17:19, 5 May 2008Thumbnail for version as of 17:19, 5 May 2008360 × 323 (20 KB)Sbyrnes321 (talk | contribs){{Information |Description=This is a demonstration that Exp(I*Pi)=-1 (called Euler's formula, or Euler's identity). It uses the formula (1+z/N)^N --> Exp(z) (as N increases). The Nth power is displayed as a repeated multiplication in the complex plane. As
16:58, 5 May 2008Thumbnail for version as of 16:58, 5 May 2008360 × 308 (18 KB)Sbyrnes321 (talk | contribs){{Information |Description=This is a demonstration that Exp(I*Pi)=-1 (called Euler's formula, or Euler's identity). It uses the formula (1+z/N)^N --> Exp(z) (as N increases). The Nth power is displayed as a repeated multiplication in the complex plane. As

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