Difference between revisions of "De Moivre's Theorem"
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| − | DeMoivre's Theorem is a very useful theorem in the mathematical fields of [[ | + | '''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 == | |
| + | This is one proof of De Moivre's theorem by [[induction]]. | ||
| + | *If <math>n>0</math>, for <math>n=1</math>, the case is obviously true. | ||
| − | == | + | :Assume true for the case <math>n=k</math>. Now, the case of <math>n=k+1</math>: |
| − | |||
| − | + | :[[Image:DeMoivreInductionP1.gif]] | |
| − | |||
| − | |||
| − | + | :Therefore, the result is true for all positive integers <math>n</math>. | |
| − | |||
| − | = | + | *If <math>n=0</math>, the formula holds true because <math>\cos(0x)+i\sin (0x)=1+i0=1</math>. Since <math>z^0=1</math>, the equation holds true. |
| − | |||
| − | + | *If <math>n<0</math>, one must consider <math>n=-m</math> when <math>m</math> is a positive integer. | |
| − | + | :[[Image:DeMoivreInductionP2.gif]] | |
| − | + | 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)^n = 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, [[Euler's identity]] states that <math>e^{ix} = \cos x+i\sin x</math>. This extends De Moivre's theorem to all <math>n\in \mathbb{R}</math>. | |
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| − | + | ==Generalization== | |
| − | |||
| − | + | [[Category:Theorems]] | |
| + | [[Category:Complex numbers]] | ||
Revision as of 05:04, 29 April 2018
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 identity states that 
. This extends De Moivre's theorem to all 
.
