MandysNotes

Friday, 13 May 2011 00:00

Euler’s Formula

By  Gideon
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The real part of \[ e^{i\alpha} \ \]

is:

\[ \text{Re}(e^{i \alpha}) = \frac{ e^{i\alpha} + e^{-i\alpha}}{2} \]

\[ =\sum_{k=0}^{\infty} (-1)^{k}\frac{\alpha^{2k}}{(2k)!} . \]

 

The imaginary part of \[ e^{i\alpha} \ \]

is:

\[ \text{Im}(e^{i\alpha}) = \frac{ e^{i\alpha} - e^{-i\alpha}}{2i} \]

\[ = \sum_{k=1}^{\infty}(-1)^{k -1}\frac{\alpha^{2k -1}}{(2k -1)!} . \]

 

We define:

\[ \text{Re}(e^{i\alpha}) = \frac{ e^{i\alpha} + e^{-i\alpha}}{2} := \cos{\alpha} ;\]

\[ \text{Im}(e^{i\alpha}) = \frac{ e^{i\alpha} - e^{-i\alpha}}{2i} := \sin{\alpha}.\]

 

Therefore:

\[ e^{i\alpha} = \text{Re} (e^{i\alpha}) + i\text{Im}(e^{i\alpha}) =\cos{\alpha} + i \sin{\alpha}. \]

 

This is Euler's Formula.

Read 3022 times Last modified on Tuesday, 01 April 2014 19:12
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