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 | + | '''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{N}</math>, <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>. |
== Proof == | == Proof == | ||
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And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math> | And thus, the formula proves true for all integral values of <math>n</math>. <math>\Box</math> | ||
− | + | Note that from the functional equation <math>f(x) = f(nx)</math> where <math>f(x) = \cos x + i\sin x</math>, we see that <math>f(x)</math> behaves like an exponential function. Indeed, by [[Euler's formula]] (<math>e^{ix} = \cos x+i\sin x\right</math>). This extends De Moivre's theorem to all <math>n\in \mathbb{R}</math>. | |
[[Category:Theorems]] | [[Category:Theorems]] | ||
[[Category:Complex numbers]] | [[Category:Complex numbers]] |
Revision as of 13:13, 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.
- Assume true for the case
. Now, the case of
:
- Therefore, the result is true for all positive integers
.
- If
, the formula holds true because
. Since
, the equation holds true.
- If
, 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, by Euler's formula ($e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. Unknown error_msg)). This extends De Moivre's theorem to all
.