MandysNotes

13 May 2011 In Complex Exponentials

Consider the exponential of a purely imaginary number:

 

\[ z = i\alpha \]

with \[ \alpha \in \mathbb{R}\ .\]

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

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

NOTRad