Euler's Identity

series[θ_]:=1+Prepend[Table[(∑_(i=1)^k)((((I θ))^i)/(i!)),{k,1,27}],0]

Manipulate[ListLinePlot[{Table[{Cos[ϕ],Sin[ϕ]},{ϕ,0,6.283185,.001}],{{0,0},{N[Re[(E^(I θ))],5],N[Im[(E^(I θ))],5]}},Table[{Re[series[θ][[j]]],Im[series[θ][[j]]]},{j,0,m,1}]},AspectRatio->Automatic,PlotRange->If[scale<0,{{N[Re[(E^(I θ))],5]-(E^scale),N[Re[(E^(I θ))],5]+(E^scale)},{N[Im[(E^(I θ))],5]-(E^scale),N[Im[(E^(I θ))],5]+(E^scale)}},{{-(E^scale),(E^scale)},{-(E^scale),(E^scale)}}],PlotStyle->{Directive[RGBColor[.25,.43,.82],Thick],Directive[RGBColor[.49,0,0],Thick],Directive[RGBColor[.6,.73,.36],Thick]},ImageSize->{400,400}],
{{θ,6.283185/2},0,6.283185,Appearance->"Labeled"},
{{m,10},1,27,1,Appearance->"Labeled"},
{{scale,1.387},-9,4},
SaveDefinitions->True]

CAPTION

Euler's identity, (e^(i π))+1=0, has been called the most beautiful equation in mathematics. It unites the most basic numbers of mathematics: e (the base of the natural logarithm), i (the imaginary unit = (Sqrt[-1]) ), π (the ratio of the circumference of a circle to its diameter), 1 (the multiplicative identity), and 0 (the additive identity) with the basic arithmetic operations: addition, multiplication and exponentiation—using each once and only once! The identity is a particular example of the more general Euler formula: (e^(i θ))=cos(θ)+i sin(θ). The series expansion of (e^z) is (∑_(n=0)^∞)((z^n)/(n!)) with partial sums (Subscript[S, m])=(∑_(n=0)^m)((z^n)/(n!)).

This Demonstration starts with Euler's identity but lets you experiment with the more general formula for various values of θ. You can see how quickly the series expansion of (e^(i θ)) converges by varying the number of terms in its partial sums with m. The green polygonal line joins the points (Subscript[S, 0]), (Subscript[S, 1]), …, (Subscript[S, m]) where z=(e^(i θ)). You can zoom in with the scale, which is logarithmic. When you zoom in sufficiently close, the center of the picture switches from the origin to the point being approximated.

DETAILS

Euler recognized that the Taylor series expansion of (e^x) when applied to an imaginary number i θ also converged and, in fact, converged to cos(θ)+i sin(θ). The terms of the series for a pure imaginary number i θ alternate between real and imaginary numbers. This provides a very interesting visualization of the convergence of the series. There are many sources for Euler's work with this formula, but [1] is the one that motivated me to create this Demonstration.

Reference

[1] R. P. Crease, The Great Equations, New York: W. W. Norton & Company, 2008 pp. 91–106.

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Euler's Identity" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/EulersIdentity/

Contributed by: Jim Kaiser

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