De Moivre's Theorem
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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. ! Missing delimiter (. inserted).)). This extends De Moivre's theorem to all .