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The Taylor series of <math> e^x </math> is | The Taylor series of <math> e^x </math> is | ||
| − | <math> e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} </math> | + | <math> e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots </math> |
Using this equation, it is possible to relate <math>e</math> to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, <math>z = a+bi</math> for example. This yields: | Using this equation, it is possible to relate <math>e</math> to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, <math>z = a+bi</math> for example. This yields: | ||
Revision as of 11:36, 2 December 2018
$ e $ and Trigonometry
The Taylor series of $ e^x $ is
$ e^x = \sum^{\infty}_{n=0}{\frac{x^n}{n!}} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots $
Using this equation, it is possible to relate $ e $ to the seemingly unrelated worlds of trigonometry and the complex numbers by simply plugging in a complex number, $ z = a+bi $ for example. This yields:
File:Eztaylor.jpg
350px
But by rearranging this, one gets the identity
File:Ezcis.jpg
350px
References:
(Reference 1)
(Reference 2)
