The derivation of the $n$th roots of a complex number is a fundamental topic in complex analysis. Let’s go through the process step by step.

Step 1: Represent the Complex Number in Polar Form

Any complex number $z$ can be written in polar form as:

$z = r(\cos \theta + i \sin \theta)$

where:

• $r = |z|$ is the modulus of the complex number.
• $\theta = \arg(z)$ is the argument (or angle) of the complex number.

Step 2: Define the Problem

We want to find the $n$th roots of $z$, meaning we need to solve the equation:

$w^n = z$

where $w$ is the unknown complex number.

Step 3: Express $w$ in Polar Form

Assume that $w$ can also be represented in polar form:

$w = \rho (\cos \phi + i \sin \phi)$

where $\rho$ is the modulus of $w$, and $\phi$ is the argument of $w$.

Step 4: Substitute and Compare

Substitute the polar form of $w$ into the equation $w^n = z$:

$(\rho (\cos \phi + i \sin \phi))^n = r(\cos \theta + i \sin \theta)$

Using De Moivre’s Theorem, which states:

$(\rho (\cos \phi + i \sin \phi))^n = \rho^n (\cos (n\phi) + i \sin (n\phi))$

we get:

$\rho^n (\cos (n\phi) + i \sin (n\phi)) = r(\cos \theta + i \sin \theta)$

Step 5: Equate Moduli and Arguments

For the equation to hold, the moduli and the arguments of both sides must be equal. Therefore, we have:

1. $\rho^n = r$ ⟹ $\rho = \sqrt[n]{r}$
2. $n\phi = \theta + 2k\pi$, where $k$ is an integer.

Step 6: Solve for $\phi$

Solve the equation for $\phi$:

$\phi = \frac{\theta + 2k\pi}{n}$

where $k$ can take values $k = 0, 1, 2, \ldots, n-1$ to get all distinct $n$th roots.

Step 7: Write the General Solution

Thus, the $n$th roots of the complex number $z$ are given by:

$w_k = \sqrt[n]{r} \left(\cos \left(\frac{\theta + 2k\pi}{n}\right) + i \sin \left(\frac{\theta + 2k\pi}{n}\right)\right)$

for $k = 0, 1, 2, \ldots, n-1$.

Summary

The $n$th roots of a complex number $z = r(\cos \theta + i \sin \theta)$ are:

$w_k = \sqrt[n]{r} \left(\cos \left(\frac{\theta + 2k\pi}{n}\right) + i \sin \left(\frac{\theta + 2k\pi}{n}\right)\right)$

These roots form the vertices of a regular $n$-sided polygon in the complex plane.

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Would you like further details or have any questions about this derivation? Here are some related questions:

1. How does De Moivre's Theorem apply to roots of unity?
2. Can you explain why the roots are equally spaced in the complex plane?
3. What is the significance of the modulus $\rho$ in the solution?
4. How would you derive the cube roots of a specific complex number?
5. What is the geometric interpretation of the $n$th roots of a complex number?
6. Can this method be extended to find roots of non-integer orders?
7. How does the principal root differ from the other roots?
8. What happens to the roots as $n$ increases?

Tip: Understanding the geometric representation of complex numbers can greatly enhance your intuition about their properties, especially when dealing with roots and powers.