Chapter 9.3, Problem 66AYU

Prove formula (6).

To determine

To show: Let $z\in C$ with $z=r\left(\cos \theta +i\sin \text{θ}\right)$. Then there exists nth of $z$ is $z_k=\sqrt[n]{\rho }\left(\cos \frac{\varphi +2\pi k}{n}+i\sin \frac{\varphi +2\pi k}{n}\right)$.

Answer to Problem 66AYU

Solution:

To find nth root of a complex number $w\ne 0$, we have to solve the equation $z^n=w$.

Let $w=\rho \left(\cos \varphi +i\sin \varphi \right)\text{and}z=r\left(\cos \theta +i\sin \text{θ}\right)$.

We can use here De Moivre's theorem, which states that if $z=r\left(\cos \theta +i\sin \theta \right)$, then,

$z^n=r^n\left(\cos n\theta +i\sin n\theta \right)=\rho \left(\cos \varphi +i\sin \varphi \right)$   -----(1)

From the above equation $r^n=\rho \text{and}n\theta =\varphi$ and we obtain the root.

$z_k=\sqrt[n]{\rho }\left(\cos \frac{\varphi }{n}+i\sin \frac{\varphi }{n}\right)$   -----(2)

Where $\sqrt[n]{\rho }$ is the positive nth root of the positive number $\rho$.

From (1) and (2) $\theta =\frac{\varphi }{n}+k\frac{2\pi }{n}$ where $k$ is an integer.

$z_k=\sqrt[n]{\rho }\left(\cos \frac{\varphi +2\pi k}{n}+i\sin \frac{\varphi +2\pi k}{n}\right)k=0,1,2\dots$.

Explanation of Solution

To find nth root of a complex number $w\ne 0$, we have to solve the equation $z^n=w$.

Let $w=\rho \left(\cos \varphi +i\sin \varphi \right)\text{and}z=r\left(\cos \theta +i\sin \text{θ}\right)$.

We can use here De Moivre's theorem, which states that if $z=r\left(\cos \theta +i\sin \theta \right)$, then,

$z^n=r^n\left(\cos n\theta +i\sin n\theta \right)=\rho \left(\cos \varphi +i\sin \varphi \right)$   -----(1)

From the above equation $r^n=\rho \text{and}n\theta =\varphi$ and we obtain the root.

$z_k=\sqrt[n]{\rho }\left(\cos \frac{\varphi }{n}+i\sin \frac{\varphi }{n}\right)$   -----(2)

Where $\sqrt[n]{\rho }$ is the positive nth root of the positive number $\rho$.

From (1) and (2) $\theta =\frac{\varphi }{n}+k\frac{2\pi }{n}$ where $k$ is an integer.

$z_k=\sqrt[n]{\rho }\left(\cos \frac{\varphi +2\pi k}{n}+i\sin \frac{\varphi +2\pi k}{n}\right)k=0,1,2\dots$.