Problem 43 - Advanced Engineering Mathematics

Question

In Problems 43 and 44 , find the principal value of the given quantity. Express answers in the form \(a+i b\). \((-1)^{(-2 i / \pi)}\)

Step-by-Step Solution

Answer

The principal value is \(e^2 + 0i\).

Step 1: Understand the Problem

The goal is to find the principal value of the expression \((-1)^{(-2i/\pi)}\), which implies finding its value in the form \(a+ib\). This involves using complex number theory and the polar form of complex numbers.

Step 2: Represent -1 in Polar Form

The number \(-1\) can be represented in polar form as \(e^{i\pi}\), since it lies on the negative real axis of the complex plane with an angle of \(\pi\) radians from the positive real axis.

Step 3: Apply the exponent

Using the polar form \((-1) = e^{i\pi}\), substitute into the expression: \[(-1)^{-\frac{2i}{\pi}} = \left(e^{i\pi}\right)^{-\frac{2i}{\pi}} = e^{i\pi \cdot \left(-\frac{2i}{\pi}\right)}\]Simplify the exponent: \[e^{\pi i \cdot \left(-\frac{2i}{\pi}\right)} = e^{2}\], because \(i^2 = -1\) and \(-\frac{2i}{\pi} \cdot i\pi = 2\).

Step 4: Express in the Form a + ib

The number \(e^2\) is a real number, having no imaginary part. Therefore, in the form \(a+ib\), \(a = e^2\) and \(b = 0\).

Step 5: Conclusion

Thus, the expression \((-1)^{(-2i/\pi)}\) simplifies to \(e^2 + 0i\). The principal value is the real number \(e^2\).

Key Concepts

Polar Form of Complex Numbers

Complex numbers are fascinating as they extend our understanding of numbers beyond the real number line. They can be expressed in a unique way known as polar form. The polar form of a complex number is very useful for calculations involving powers and roots.

In polar form, a complex number is expressed as a magnitude and an angle. Instead of representing a complex number as \(a+bi\), we express it as \(re^{i\theta}\). Here, \(r\) is the magnitude or modulus of the complex number, and \(\theta\) is the angle, known as the argument. The angle \(\theta\) indicates how far the number is from the positive real axis on the complex plane.

This form is particularly useful because it simplifies multiplication and division of complex numbers. When raising a complex number to a power, as in our exercise, the polar form allows you to easily compute powers by multiplying the exponents.

Euler's Formula

Euler's formula is a beautiful equation in complex analysis and is central to the polar form of complex numbers. It states that for any real number \(\theta\), the equation \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \) holds true.

Euler's formula seamlessly combines exponential, trigonometric, and complex functions into one elegant expression. This relationship allows us to convert back and forth between exponential and trigonometric representations of complex numbers. In the original exercise, the formula is utilized to express \(-1\) as \(e^{i\pi}\). This representation is especially useful because it allows us to handle powers and logarithms of complex numbers easily.

Using Euler's formula makes the process of solving complex equations, such as finding a principal value, straightforward by leveraging the properties of exponential functions.

Principal Value of Complex Expressions

The principal value in the context of complex numbers is a way to find a particular value of a multi-valued function, like \(z^w\) in complex analysis. Particularly, when dealing with expressions like \((-1)^{(-2i/\pi)}\), it's vital to determine a single, representative value.

For complex numbers, a function like exponentiation can have multiple values due to the periodicity of the argument in its polar form. For instance, the angle component \(\theta\) can differ by integer multiples of \(2\pi\). Thus, there are infinite results, but the principal value is the one with the smallest positive angle. This ensures a consistent and universally accepted answer.

In the step-by-step solution for this problem, recognizing \(-1\) as \(e^{i\pi}\) helped in finding a straightforward principal value by reducing the expression to an easily computable form, ultimately yielding the real number \(e^2\).