De Movire's Theorem: Applications for Root Extraction

[revitgaur]

I want to know about applications of De Movire's theorem for root extraction.

 

[QuantumQuest]

Perhaps the greatest value of De Moivre's Theorem, is the ability to find the n distinct roots of a complex number.

Let $z = p(cos\theta + isin\theta)$ and let $z^{n} = w$. Then if $w = r(cos\phi+ isin\phi)$, $z^{n} = [p(cos\theta + isin\theta)]^{n}$ we have
that $p^{n}[cos(n\theta) +isin(n\theta)] = r(cos\phi+ isin\phi)$. That implies that $p^{n} = r$ and $n\theta = \phi$, or equivalently $p = \sqrt[\leftroot{-2}\uproot{2}n]{r}$ and $\theta = \frac{\phi}{n}$. But $sin$ and $cos$ have period of $2\pi$ so $n\theta = \phi + 2\pi k$ or equivalently $\theta = \frac{\phi + 2\pi k}{n}$, $k = 0, 1, 2, \cdots, n - 1$. If we set $k = n$ the solutions are repeated . So, for a positive integer $n$, we find $n$ distinct $n-th$ roots for the complex number $w = r(cos\phi+ isin\phi)$ : $z = \sqrt[\leftroot{-2}\uproot{2}n]{r} [cos \frac{\phi + 2\pi k}{n} + isin \frac{\phi + 2\pi k}{n}]$.

With the help of the above formula, we can compute the $n- th$ roots of e.g $1$ or maybe some specific roots, e.g $4 - th$.

 

[revitgaur]

Perhaps the greatest value of De Moivre's Theorem is the ability to find the n distinct roots of a complex number.

Let $z = p(cos\theta + isin\theta)$ and let $z^{n} = w$. Then if $w = r(cos\phi+ isin\phi)$, $z^{n} = [p(cos\theta + isin\theta)]^{n}$ we have
that $p^{n}[cos(n\theta) +isin(n\theta)] = r(cos\phi+ isin\phi)$. That implies that $p^{n} = r$ and $n\theta = \phi$, or equivalently $p = \sqrt[\leftroot{-2}\uproot{2}n]{r}$ and $\theta = \frac{\phi}{n}$. But $sin$ and $cos$ have period of $2\pi$ so $n\theta = \phi + 2\pi k$ or equivalently $\theta = \frac{\phi + 2\pi k}{n}$, $k = 0, 1, 2, \cdots, n - 1$. If we set $k = n$ the solutions are repeated . So, for a positive integer $n$, we find $n$ distinct $n-th$ roots for the complex number $w = r(cos\phi+ isin\phi)$ : $z = \sqrt[\leftroot{-2}\uproot{2}n]{r} [cos \frac{\phi + 2\pi k}{n} + isin \frac{\phi + 2\pi k}{n}]$.

With the help of the above formula we can compute the $n- th$ roots of e,g $1$ or maybe some specific roots e.g $4 - th$.

Thanks for helping me