Difference between revisions of "De Moivre's Theorem"
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{{WotWAnnounce|week=September 5- September 11}} | {{WotWAnnounce|week=September 5- September 11}} | ||
| − | '''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar form]] to be easily raised to certain powers. It states that for <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{ | + | '''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex numbers]]. It allows complex numbers in [[polar form]] to be easily raised to certain powers. It states that for <math>x\in\mathbb{R}</math> and <math>n\in\mathbb{Z}</math>, <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>. |
== Proof == | == Proof == | ||
Revision as of 12:15, 10 September 2008
| This is an AoPSWiki Word of the Week for September 5- September 11 |
DeMoivre's Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for 
and 
, 
.
Proof
This is one proof of De Moivre's theorem by induction.
- If
, for 
, the case is obviously true.
, for 
, the case is obviously true.- Assume true for the case
. Now, the case of 
:
. Now, the case of 
:
- Therefore, the result is true for all positive integers
.
- If
, the formula holds true because 
. Since 
, the equation holds true.
, the formula holds true because 
. Since 
, the equation holds true.- If
, one must consider 
when 
is a positive integer.
, one must consider 
when 
is a positive integer.
And thus, the formula proves true for all integral values of 
. 
Note that from the functional equation 
where 
, we see that 
behaves like an exponential function. Indeed, Euler's formula states that $e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. Unknown error_msg). This extends De Moivre's theorem to all 
.
