Euler's Identity
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
Euler's identity, 
, has been called the most beautiful equation in mathematics. It unites the most basic numbers of mathematics: 
(the base of the natural logarithm), 
(the imaginary unit = 
), 
(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: 
. The series expansion of 
is 
with partial sums 
.
Contributed by: Jim Kaiser (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Euler recognized that the Taylor series expansion of 
when applied to an imaginary number 
also converged and, in fact, converged to 
. The terms of the series for a pure imaginary number 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.
Permanent Citation
Euler Product for the Zeta Function
S. M. Blinder
The Argument Principle
Andrzej Kozlowski
The Geometry of Integrating a Power around the Origin
John Custy
Complex Power Plot
Sue Hurley
Riemann Surface Transition
Michael Trott
Euler Transformation
Eliot Ball
Frieze Functions
Izidor Hafner
Partial Sums of a Lambert Series
Enrique Zeleny
Stolz Angle
Andrzej Kozlowski
Convergence of a Power Series for Polylogarithm
Andrzej Kozlowski
