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 an $\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)$

Proof

This is one proof of De Moivre's theorem by induction.

  • If $n>0$, for $n=1$, the case is obviously true.
Assume true for the case $n=k$. Now, the case of $n=k+1$:
DeMoivreInductionP1.gif
Therefore, the result is true for all positive integers $n$.
  • If $n=0$, the formula holds true because $\cos(0x)+i\sin (0x)=1+i0=1$. Since $z^0=1$, the equation holds true.
  • If $n<0$, one must consider $n=-m$ when $m$ is a positive integer.
DeMoivreInductionP2.gif

And thus, the formula proves true for all integral values of $n$. $\Box$

By Euler's formula ($e^{ix} = \cos x+i\sin x\right$ (Error compiling LaTeX. ! Missing delimiter (. inserted).)), this can be extended to all real numbers $n$.

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