In §
2.10, De Moivre's theorem was introduced as a consequence of
Euler's identity:

To provide some further insight into the ``mechanics'' of Euler's
identity, we'll provide here a direct proof of De Moivre's theorem for
integer 
using
mathematical induction and elementary trigonometric
identities.
Proof:
To establish the ``basis'' of our mathematical
induction proof, we may
simply observe that De Moivre's theorem is trivially true for

. Now assume that De Moivre's theorem is true for some positive
integer

. Then we must show that this implies it is also true for

,
i.e.,
![$\displaystyle \left[\cos(\theta) + j \sin(\theta)\right] ^{n+1} = \cos[(n+1)\theta] + j \sin[(n+1)\theta]. \protect$](http://www.dsprelated.com/josimages_new/mdft/img351.png) |
(3.2) |
Since it is true by hypothesis that
multiplying both sides by
![$ [\cos(\theta) + j \sin(\theta)]$](http://www.dsprelated.com/josimages_new/mdft/img353.png)
yields
From
trigonometry, we have the following
sum-of-angle identities:
These identities can be proved using only arguments from classical
geometry.
3.8Applying these to the right-hand side of Eq.

(
3.3), with

and

, gives Eq.

(
3.2), and
so the induction step is proved.

De Moivre's theorem establishes that integer powers of
![$ [\cos(\theta) + j \sin(\theta)]$](http://www.dsprelated.com/josimages_new/mdft/img353.png)
lie on a circle of radius 1 (since

, for all
![$ \phi\in[-\pi,\pi]$](http://www.dsprelated.com/josimages_new/mdft/img364.png)
). It
therefore can be used to determine all

of the

th
roots of unity
(see §
3.12 above).
However, no definition of

emerges readily from De Moivre's
theorem, nor does it establish a definition for
imaginary exponents
(which we defined using
Taylor series expansion in §
3.7 above).
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