# MandysNotes

Friday, 13 May 2011 23:50

## The Exponential of a Purely Imaginary Number

By  Gideon
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Consider the exponential of a purely imaginary number:

$z = i\alpha$

with $\alpha \in \mathbb{R}\ .$

$e^{i\alpha} = \lim_{n \to \infty } (1 + \frac{i\alpha}{n})^{n}$

$= \sum_{k=0}^{\infty }\frac{i\alpha^{k}}{k!} .$

Note that:

$| e^{i\alpha} |^{2} = e^{i\alpha} e^{-i\alpha}$

$= e^{ i\alpha - i\alpha} = e^{0} = 1.$

So for every $\alpha \in \mathbb{R}\,$

$e^{i\alpha}$

is a complex number with modulus 1.

We can think of $e^{i\alpha} \$as a unit vector in the complex plane.