Problem 48 - Advanced Engineering Mathematics

Question

The laws of exponents hold for complex numbers \(\alpha\) and \(\beta\) : \(z^{\alpha} z^{\beta}=z^{\alpha+\beta}, \quad \frac{z^{\alpha}}{z^{\beta}}=z^{\alpha-\beta}, \quad\left(z^{\alpha}\right)^{n}=z^{n \alpha}, \quad n\) an integer However, the last law is not valid if \(n\) is a complex number. Verify that \(\left(i^{i}\right)^{2}=i^{2 i}\), but \(\left(i^{2}\right)^{i} \neq i^{2 i}\)

Step-by-Step Solution

Answer

Both expressions are equal; previous equations are consistent with: \((i^i)^2 = i^{2i}\) and \((i^2)^i = i^{2i}\).

Step 1: Understand the Problem

We need to calculate two expressions involving complex exponentiation and verify if they are equal to another common power. Specifically, verify if \((i^i)^2\) equals \(i^{2i}\) and if \((i^2)^i\) equals \(i^{2i}\).

Step 2: Calculate \((i^i)^2\)

First, calculate \(i^i\). Remember that for any complex number in exponential form, we use the Euler's formula: \(i = e^{i\frac{\pi}{2}}\). Thus, \(i^i = (e^{i\frac{\pi}{2}})^i = e^{-\frac{\pi}{2}}\). Next, calculate \((i^i)^2 = (e^{-\frac{\pi}{2}})^2 = e^{-rac{\pi}{2} \times 2} = e^{-rac{\pi}{2} \times 2} = e^{-\pi}\).

Step 3: Calculate \(i^{2i}\)

Use the formula from Step 2: \(i = e^{i\frac{\pi}{2}}\). Thus, \(i^{2i} = (e^{i\frac{\pi}{2}})^{2i} = e^{i\frac{\pi}{2} \times 2i} = e^{-\pi}\).

Step 4: Compare \((i^i)^2\) and \(i^{2i}\)

From Steps 2 and 3, we found that \((i^i)^2 = e^{-\pi}\) and \(i^{2i} = e^{-\pi}\). These are equal.

Step 5: Calculate \((i^2)^i\)

Recognize that \(i^2 = -1\). Next, find \((-1)^i\) using the exponential form: \(-1 = e^{i\pi}\) implies \((-1)^i = (e^{i\pi})^i = e^{-\pi}\). Thus, \((i^2)^i = e^{-\pi}\).

Step 6: Calculate \(i^{2i}\) again for comparison

We've already calculated \(i^{2i} = e^{-\pi}\) in Step 3. However, if we were strictly following the distinct steps, let's assure the value stays consistent with past calculations.

Step 7: Compare \((i^2)^i\) and \(i^{2i}\)

From Step 5, we have \((i^2)^i = e^{-\pi}\). From Step 6, we have \(i^{2i} = e^{-\pi}\). We find that \((i^2)^i = i^{2i}\), which means we need to reevaluate the question as both expressions are calculated to be equal.

Key Concepts

Complex Numbers

Complex numbers are fascinating and form an essential part of mathematical theory with vast applications in various fields such as engineering and physics. A complex number is expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(b\) being a real number and \(i\) being the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).

When working with complex numbers, particularly in exponentiation, it’s crucial to understand their representation on the complex plane. Here, the horizontal axis represents the real part, while the vertical axis represents the imaginary part. A point on the plane corresponds to the complex number \(a + bi\).

Complex numbers can also be represented in polar form as \(r(\cos\theta + i\sin\theta)\), where \(r\) is the magnitude, and \(\theta\) is the argument of the complex number. This form links beautifully to Euler's formula, bridging complex numbers and exponential functions.

Euler's Formula

Euler's formula establishes a profound connection between trigonometry and complex exponentials. It states that for any real number \(\theta\), \(e^{i\theta} = \cos\theta + i\sin\theta\). This formula is incredibly powerful as it allows complex numbers to be expressed as exponentials, which is especially useful in computations involving roots and powers.

Euler's formula also shines when working with complex exponentiation. For instance, to express \(i\) in exponential form, recognize that \(i = e^{i\frac{\pi}{2}}\). This is derived by setting \(\theta = \frac{\pi}{2}\) in Euler's formula, yielding \(e^{i\frac{\pi}{2}} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = i\).

This relationship facilitates calculations like \(i^i\), as shown in the exercise. Such conversions allow for a straightforward manipulation of expressions using familiar rules of exponentials, vastly simplifying complex computations.

Exponentiation Laws

Exponentiation laws simplify many problems by providing rules for handling powers and roots. When dealing with real numbers, these rules are straightforward, but they hold good for complex numbers as well, with some caveats.

The basic laws are:

• \(z^{\alpha} z^{\beta} = z^{\alpha + \beta}\)
• \(\frac{z^{\alpha}}{z^{\beta}} = z^{\alpha - \beta}\)
• \((z^{\alpha})^n = z^{n\alpha}\) for integer \(n\)

A critical point addressed in the exercise is the exception in the last law when \(n\) is a complex number. The results don't necessarily conform to real-number expectations.

For example, we see in the exercise, while \((i^i)^2\) and \(i^{2i}\) result in the same value, \((i^2)^i\) is not the same computation and generally hold different results under further examination. Such situations underline the importance of understanding when certain rules do not hold in complex scenarios. Mastering these nuanced rules enables one to adeptly navigate complex number problems.