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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.

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