Summary

DescriptionEulerFormulaAsLimit.gif	English: This is a demonstration of Euler's formula, exp(i*pi/3) = cos(pi/3) + i*sin(pi/3). 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 the point on the unit circle of the complex plane (dashed line) at angle pi/3 radians from +1. Variant of File:ExpIPi.gif.
Date	22 January 2012
Source	Own work   This diagram was created with Mathematica.
Author	Sbyrnes321

(* Source code written in Mathematica 6.0, by Steve Byrnes, 2012. I release this code into the public domain. *)

frameticks = {-.25, 0, .25, .5, .75, 1, 1.25};
plot1 = Table[
  Show[ListPlot[
    Table[{Re[(1 + I*Pi/(3 n))^m], Im[(1 + I*Pi/(3 n))^m]}, {m, 0, n}],
    Joined -> True, PlotMarkers -> Automatic, 
    PlotRange -> {{-.3, 1.1}, {-.3, 1.1}}, AxesOrigin -> {0, 0}, 
    Frame -> True, FrameLabel -> {"Real part", "Imaginary part"}, 
    FrameTicks -> {{frameticks, None}, {frameticks, None}}, 
    Frame -> True, PlotLabel -> "n = " <> ToString[n], 
    LabelStyle -> Medium, AspectRatio -> Automatic], 
   ParametricPlot[{Sin[t], Cos[t]}, {t, 0, 2*Pi}, 
    PlotStyle -> Directive[Gray, Dashed]]],
  {n, {1, 2, 3, 4, 5, 10, 20, 50, 100}}];
SetDirectory["C:\\Users\\Steve\\Desktop"];
Export["test.gif", plot1, "DisplayDurations" -> {2}, "AnimationRepititions" -> Infinity, ImageSize -> 200];

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.

The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse