# De Moivre's Theorem

**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 $e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. ! Missing delimiter (. inserted).). This extends De Moivre's theorem to all .