Question

Evaluate each of the following in \(x+i y\) form, and compare with a computer solution. $$\left(\frac{1+i \sqrt{3}}{2}\right)^{i}$$

Short answer

The result is \(e^{-\pi/3}\).

Step-by-step solution

- Rewrite the Complex Number

Express the complex number in polar form. The given complex number is \(\frac{1+i \sqrt{3}}{2}\).

- Convert to Polar Form

To convert \(\frac{1+i \sqrt{3}}{2}\) to polar form, find the magnitude \(r\) and the argument \(\theta\). The magnitude is \(|z| = \sqrt{x^2 + y^2} = \sqrt{ \left( \frac{1}{2} \right)^2 + \left( \frac{\sqrt{3}}{2} \right)^2 } = 1\) due to the properties of a unit circle. For \(\theta\), use \(\tan^{-1}(y/x)\). Here, \(x = \frac{1}{2}\) and \(y = \frac{\sqrt{3}}{2}\), leading to \(\theta = 60° = \pi/3\). Thus, in polar form, the complex number is \(z = e^{i \pi/3}\).

- Exponentiation

Use the identity \( (e^{i \theta})^i = e^{-\theta}\). Here, \(z^i = (e^{i \pi/3})^i = e^{-\pi/3}\).

- Conversion to Rectangular Form

Convert \(e^{-\pi/3}\) back to rectangular form. Since it is a purely real number, the rectangular form is: \(e^{-\pi/3} + 0i\).

Key concepts

Polar to Rectangular Conversion

Complex numbers can be expressed in both polar and rectangular forms. The rectangular form of a complex number is written as: \( x + iy \), where \( x \) and \( y \) are real numbers. On the other hand, the polar form represents a complex number as: \( r e^{i \theta} \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle).

To convert a complex number from polar to rectangular form, use the following formulas:

• \( x = r \cos{\theta} \)
• \( y = r \sin{\theta} \)

For instance, if we have a complex number in polar form \( 1 e^{i\pi/3} \), the conversion to rectangular form can be done by:

• \( x = 1 \cos{\pi/3} = \frac{1}{2} \)
• \( y = 1 \sin{\pi/3} = \frac{\sqrt{3}}{2} \)

This results in the rectangular form of: \( \frac{1}{2} + i \frac{\sqrt{3}}{2} \).

This method allows us to move back and forth between different representations, depending on which is more convenient for solving a problem.

Complex Exponentiation

Complex exponentiation involves raising a complex number to a power, which can also be a complex number. This concept makes use of polar form for simplification. Suppose we need to evaluate \( z^i \) where \( z = \frac{1 + i\sqrt{3}}{2} \).

First, convert \( z \) to its polar form. As per the earlier steps, this gives us: \( z = 1 e^{i\pi/3} \). Now, applying the exponentiation:

• \( z^i = (1 e^{i \pi/3})^i \)
• Use the identity \( (e^{i \theta})^i = e^{-\theta} \), which simplifies our expression to:
• \( e^{-\pi/3} \)

Since \( e^{-\pi/3} \) is a purely real number, the result in rectangular form is just \( e^{-\pi/3} + 0i \).

Here, complex exponentiation demonstrates how it leverages polar form to simplify calculations that might otherwise be cumbersome.

Euler's Formula

Euler's formula is a fundamental formula in complex analysis that relates complex exponentiation with trigonometric functions. It states:

• \( e^{i\theta} = \cos{\theta} + i\sin{\theta} \)

This formula plays a crucial role in the conversion between rectangular and polar forms of complex numbers.

In this exercise, Euler's formula helped convert the given complex number: \( \frac{1 + i\sqrt{3}}{2} \) from rectangular to polar form. Using Euler's formula, we can express this complex number as:

• \( e^{i\pi/3} = \cos{\pi/3} + i\sin{\pi/3} \)
• \( e^{i\pi/3} = \frac{1}{2} + i \frac{\sqrt{3}}{2} \)

This equation provides a straightforward way to see how much a complex number spins around the origin (\( \theta \)), and how far it is from the origin (\( r \)), making complex calculations considerably more manageable.