Question

The \(N+1\) complex numbers \(\omega_{m}\) are given by \(\omega_{m}=\exp (2 \pi i m / N)\) for \(m=\) \(0,1,2, \ldots, N\) (a) Evaluate the following: (i) \(\sum_{m=0}^{N} \omega_{m}\), (ii) \(\sum_{m=0}^{N} \omega_{m}^{2}\), (iii) \(\sum_{m=0}^{N} \omega_{m} x^{m}\). (b) Use these results to evaluate (i) \(\sum_{m=0}^{N}\left[\cos \left(\frac{2 \pi m}{N}\right)-\cos \left(\frac{4 \pi m}{N}\right)\right]\) (ii) \(\sum_{m=0}^{3} 2^{m} \sin \left(\frac{2 \pi m}{3}\right)\).

Short answer

(i) 0, (ii) 0, (iii) 0

Step-by-step solution

- Definition of Complex Number

The complex numbers are where m=0,1,2,…,N.

- Summation of Complex Numbers

We need to evaluate the sum and note that where m goes from 0 to N.

- Finding First Summation

Recognize that is the roots of unity, hence the sum of all equals zero: .

- Finding Third Summation

and evaluating the sum gives .

- Evaluate First Expression in Part (b)

Using results from (a), the summation results are: .

- Evaluate Second Expression in Part (b)

thus, .

Key concepts

Roots of Unity

In the realm of complex numbers, 'roots of unity' play a significant role. These are special complex numbers that, when raised to a certain power (usually denoted by N), equal one. Mathematically, they are represented as \(\text{exp}\frac{2\text{π}im}{N}\), for m = 0, 1, 2, …, N. These roots generate a symmetrical distribution of points on the unit circle. Here are a few important aspects:

• The collection \(\text{ω}_m\) forms an N-dimensional vector space over the complex numbers.
• For any integer k, the N-th roots of unity satisfy \(\text{ω}_m^N=1\).
• The sum of all roots of unity is zero, \(\text{∑}_{m=0}^N \text{ω}_m =0\).

Understanding these basics helps in solving complex summations and grasping their symmetric properties.

Complex Exponential Function

The complex exponential function connects exponential functions with trigonometry, offering a powerful tool in physics and engineering. It is defined as \(\text{exp}(iθ) = \text{cos}(θ) + i \text{sin}(θ)\). This representation helps in converting between rectangular and polar coordinates in the complex plane.

• For an angle θ, the complex exponential function \(\text{exp}(iθ)\) maps to a point on the unit circle with coordinates (cos(θ), sin(θ)).
• This function adheres to Euler's formula, \(\text{e}^{iθ}\) = \(\text{cos(θ) + i sin(θ)}\).

The given problem uses this to establish the values of \(ω_m\), simplifying complex trigonometric relationships into a more digestible form.

Summation of Complex Numbers

Summation of complex numbers involves adding real parts and imaginary parts separately. In the exercise, we're concerned with sums of specific complex sequences:

• The sum \(∑_{m=0}^{N}ω_m\) leverages the property that the sum of all N-th roots of unity equals zero.
• For other sums like \(\text{∑}_{m=0}^Nω_m^2\), similar properties can simplify calculations, noting rotational symmetries.
• When summing series like \(\text{∑}_{m=0}^{N}ω_mx^m\), recognizing geometric series forms provides a quick resolution.

This knowledge aids in evaluating complex sequences effortlessly, ensuring precision in both real and imaginary summations.

Trigonometric Identities in Summation

Trigonometric identities simplify summations, especially those involving sine and cosine functions. The following identities are pivotal:

• The identity \( \text{cos}(A) - \text{cos}(B) = -2\text{sin}\frac{A+B}{2}\text{sin}\frac{A-B}{2}\) helps break down complex expressions.
• For summing series like \(\text{∑}_{m=0}^{N}[\text{cos}\frac{2\text{π}m}{N} - \text{cos}\frac{4\text{π}m}{N}]\), trigonometric identities reduce the problem to simpler basic components.
• Similarly, examining \( \text{∑}_{m=0}^{N}2^m\text{sin}\frac{2\text{π}m}{3}\) through these identities simplifies the analysis and ensures correct summation.

Mastering these identities can demystify complex summations, making them more approachable and manageable.