Problem 8 - Complex Analysis

Question

Existence of \(n^{\text {th }}\) roots Assume \(a \in \mathbb{C}\) and \(n \in \mathbb{N} .\) A complex number \(z\) is called (an) \(n^{\text {th }}\) root of \(a\) if \(z^{n}=a\) Show: If \(a=r(\cos \varphi+\mathrm{i} \sin \varphi) \neq 0\), then \(a\) has exactly \(n\) (different) \(n^{\text {th }}\) roots, namely the complex numbers $$ z_{\nu}=\sqrt[n]{r}\left(\cos \frac{\varphi+2 \pi \nu}{n}+\mathrm{i} \sin \frac{\varphi+2 \pi \nu}{n}\right), \quad 0 \leq \nu \leq n-1 $$ In the special case \(a=1\) (thus \(r=1, \varphi=0)\), we get Theorem I.1.7.

Step-by-Step Solution

Answer

For nonzero \( a \), there are exactly \( n \) distinct \( n^{\text{th}} \) roots, given by specific formulas for \( z_\nu \).

Step 1: Representation of Complex Numbers

A complex number \( a \) can be expressed in polar form as \( a = r (\cos \varphi + i \sin \varphi) \), where \( r = |a| \) is the modulus of \( a \) and \( \varphi \) is the argument of \( a \). This representation is convenient for finding roots.

Step 2: Definition of nth Root

An \( n^{\text{th}} \) root \( z \) of \( a \) satisfies \( z^n = a \). In polar form, \( z \) will have modulus \( \sqrt[n]{r} \) and we need to determine its argument.

Step 3: Argument of the nth Root

For \( z \), we choose its argument to be \( \frac{\varphi + 2\pi u}{n} \), where \( u \in \{0, 1, 2, \ldots, n-1\} \). The term \( 2\pi u \) accounts for the periodic nature of arguments in the complex plane.

Step 4: Construction of nth Roots

We construct \( n \) distinct roots, \( z_{u} = \sqrt[n]{r} \left( \cos \frac{\varphi + 2\pi u}{n} + i \sin \frac{\varphi + 2\pi u}{n} \right) \). Each root corresponds to different values of \( u \).

Step 5: Verify Number of Distinct Roots

Since \( 0 \leq u \leq n-1 \), we have exactly \( n \) distinct \( n^{\text{th}} \) roots. The different choices for \( u \) ensure that each root has a unique argument, all residing at different angles from one another at regular intervals in the complex plane.

Step 6: Special Case \(a=1\)

If \( a = 1 \), then \( r = 1 \) and \( \varphi = 0 \). Substituting these values into the formula gives \( z_{u} = \cos \frac{2\pi u}{n} + i \sin \frac{2\pi u}{n} \), the \( n^{\text{th}} \) roots of unity.

Key Concepts

nth Roots

The concept of nth roots in the complex plane revolves around finding a number that, when raised to the power of n, results in the given complex number. A complex number can be expressed in polar form, which makes it easier to find its roots.
First, consider a complex number \(a\) represented as \(a = r(\cos \varphi + i \sin \varphi)\).
This formulation allows us to determine its nth roots by considering the modulus \(r\) and the argument \(\varphi\).

• The modulus \(\sqrt[n]{r}\) is the magnitude of the nth root.
• The argument is \(\frac{\varphi + 2\pi u}{n}\) where \(u\) ranges from 0 to \(n-1\).

This formulation implies there are exactly \(n\) distinct nth roots corresponding to different values of \(u\) underlining the periodic nature of angles in the unit circle.

Polar Representation

The polar representation of complex numbers is a powerful tool used to navigate the complex plane. Each complex number can be represented as \(a = r (\cos \varphi + i \sin \varphi)\), where:

• \(r\) is the modulus or the absolute value, indicating how far the number is from the origin.
• \(\varphi\) is the argument, representing the angle made with the positive x-axis.

In polar form, multiplication and finding powers become easier than in rectangular form.
This representation aids significantly in calculating the nth roots by providing a straightforward way to express and manipulate the angles and magnitudes of complex numbers.

Complex Plane

The complex plane is an extension of the standard two-dimensional coordinate system used to represent complex numbers geometrically. In this plane:

• The horizontal axis represents the real part.
• The vertical axis represents the imaginary part.

Each point on this plane corresponds to a complex number \(a + bi\).
The advantage of the complex plane lies in its ability to clearly depict concepts like convergence, periodicity, and especially rotation, all of which are fundamental when calculating and visualizing nth roots and their arguments.
Visualizing nth roots as points spaced evenly around the unit circle provides a better understanding of their properties.

Roots of Unity

Roots of unity are special cases of nth roots where the modulus \(r = 1\).
These correspond to the solutions of the equation \(z^n = 1\) in the complex plane.
For each nth root of unity \(z_{u}\), the formula simplifies to:
\[ z_{u} = \cos \frac{2\pi u}{n} + i \sin \frac{2\pi u}{n} \]
which represents points evenly distributed on the unit circle.
The beauty of roots of unity lies in their symmetrical arrangement around the circle, a feature that simplifies many problems in mathematics such as polynomial factorization and Fourier transforms.