# Difference between revisions of "De Moivre's Theorem"

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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 formula]] 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>. | 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 formula]] 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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[[Category:Theorems]] | [[Category:Theorems]] | ||

[[Category:Complex numbers]] | [[Category:Complex numbers]] |

## Revision as of 22:28, 27 March 2017

**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, Euler's formula states that . This extends De Moivre's theorem to all .