Problem 13 - Complex Analysis for Mathematics and Engineering

Question

Let \(z_{k} \neq 1\) be an \(n\) th root of unity. Prove that \(1+z_{k}+z_{k}^{2}+\cdots+z_{k}^{n-1}=0\).

Step-by-Step Solution

Answer

The sum of the nth roots of unity (excluding 1) is zero: \(1+z_k+z_k^2+\cdots+z_k^{n-1}=0\).

Step 1: Understand the Definition of nth Root of Unity

An \(n\)th root of unity is a complex number \(z_k\) such that \(z_k^n = 1\). The different \(n\)th roots of unity are \(1, \omega, \omega^2, \ldots, \omega^{n-1}\) where \(\omega = e^{2\pi i / n}\). Each root corresponds to different powers of \(\omega\).

Step 2: Form the Sum of Powers of z_k

Consider the series \(1 + z_k + z_k^2 + \cdots + z_k^{n-1}\). This is a geometric series with \(n\) terms, first term \(a = 1\), and common ratio \(r = z_k\).

Step 3: Apply the Geometric Series Sum Formula

The sum of a geometric series is given by the formula: \[ S_n = \frac{a(1-r^n)}{1-r} \]Here, substitute \(a = 1\) and \(r = z_k\), giving:\[ S_n = \frac{1(1 - z_k^n)}{1 - z_k} \].

Step 4: Simplify Using the Property of z_k

Since \(z_k\) is an \(n\)th root of unity, by definition, \(z_k^n = 1\). Substitute this into the sum formula:\[ S_n = \frac{1 - 1}{1 - z_k} = \frac{0}{1 - z_k} = 0 \].

Step 5: Conclusion on the Sum

As calculated, \(1 + z_k + z_k^2 + \cdots + z_k^{n-1} = 0\) for any \(n\)th root of unity where \(z_k eq 1\). This result depends on the property that the powers of roots of unity sum to zero except for the entire cycle including \(z_k=1\).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This series takes the form:

• First term: \(a\)
• Common ratio: \(r\)

A geometric series with \(n\) terms can be summed using the formula: \[S_n = \frac{a(1 - r^n)}{1 - r}\]where:- \(S_n\) is the sum of the series,- \(a\) is the first term, and- \(r\) is the common ratio.In the context of the nth roots of unity, this formula is particularly useful because the powers of a root of unity form a geometric series. The sum evaluates to zero, except when the series includes all powers encompassing the full circle back to 1 (i.e., the starting element in the sequence). This property is what makes the formula so powerful in simplifying problems involving roots of unity.

Complex Number

Complex numbers expand our number system to include solutions to equations like \(x^2 + 1 = 0\). They are written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

• Real part: \(a\)
• Imaginary part: \(b\)
• Imaginary unit \(i\): Represents the square root of -1, fulfilling \(i^2 = -1\)

Complex numbers are essential for understanding nth roots of unity. The unit circle on the complex plane helps in visualizing these roots, as each nth root is a vertex of a regular polygon inscribed in the unit circle, equally spaced. This geometric interpretation aids comprehension of operations with roots of unity, such as multiplication and exponentiation. Complex numbers enable a more flexible number system and allow us to work with concepts that are "complex" by nature, such as those involving rotation in the plane.

Sum of Roots of Unity

The sum of the nth roots of unity (excluding the root which equals 1) is zero. This feature arises from their symmetric arrangement on the complex plane.Consider the nth roots of unity: These can be represented as powers of \(\omega\), where \(\omega = e^{2\pi i / n}\), forming the sequence:

• 1
• \(\omega\)
• \(\omega^2\)
• \(\cdots\)
• \(\omega^{n-1}\)

The geometric symmetry around the origin ensures that when these are summed, the result is zero. Mathematically, if we consider a geometric series where \(z_k eq 1\), the sum is:\[1 + \omega + \omega^2 + \cdots + \omega^{n-1} = 0\]This outcome depends on the property \(z_k^n = 1\), a defining characteristic of roots of unity. In advanced applications, understanding this property helps in computing Fourier transforms and other fields where symmetry and periodicity play critical roles. The zero sum reflects the delicate balance of angles associated with the roots of unity, reinforcing the concept of symmetry in complex numbers.