Problem 40 - Advanced Engineering Mathematics

Question

In Problems 39-42, find all values of the given quantity. $$ 3^{i / \pi} $$

Step-by-Step Solution

Answer

The values are \(\cos((\ln(3)/\pi) + 2n\pi) + i\sin((\ln(3)/\pi) + 2n\pi)\) for integer \(n\).

Step 1: Understand the expression

We start with the expression given: \(3^{i / \pi}\). This expression involves a complex exponent. The goal is to convert it into a form that we can easily evaluate to find all possible values.

Step 2: Use Euler's Formula and Conversion

We know that Euler's formula involves complex exponentials: \(e^{ix} = \cos x + i\sin x\). The expression \(b^{z}\), where \(z\) is a complex number, can be written as \(e^{z\ln(b)}\). Therefore, \(3^{i/\pi}\) can be rewritten as \(e^{i/\pi \cdot \ln(3)}\).

Step 3: Apply Euler's Formula

Using Euler's formula, \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\), we find \(\theta\) in our context: \(e^{i \cdot (\ln(3)/\pi)} = \cos(\ln(3)/\pi) + i\sin(\ln(3)/\pi)\). This provides the value of the expression.

Step 4: Recognize the periodic nature

Since \(e^{i\theta}\) is periodic with period \(2\pi\), the expression has infinitely many values of the form \(\cos((\ln(3)/\pi) + 2n\pi) + i\sin((\ln(3)/\pi) + 2n\pi)\) for any integer \(n\). Every value corresponds to a unique integer \(n\).

Key Concepts

Euler's Formula

Euler's formula is a beautiful connection between trigonometry and complex exponentials. It states that for any real number \( x \), the expression \( e^{ix} = \cos x + i\sin x \) holds true. This formula bridges the exponential function with sine and cosine, offering a powerful tool to simplify calculations involving complex numbers.

When dealing with expressions like \( b^z \) where \( z \) is a complex number, Euler's formula can be used to transform complex exponentiation into a more manageable form. By expressing \( z \) in terms of real and imaginary parts, one can leverage Euler’s formula to unlock the trigonometric components embedded within.

This transformation not only simplifies solving but also aids in visualizing complex numbers as points or vectors on a two-dimensional plane, representing rotations and oscillations.

Complex Numbers

Complex numbers are an extension of the real numbers and are essential in understanding many mathematical concepts. They are composed of a real part and an imaginary part, typically written as \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

• The real part \( a \) corresponds to the horizontal axis in the complex plane, often referred to as the real axis.
• The imaginary part \( b \) aligns with the vertical axis, known as the imaginary axis.

Complex numbers allow for solutions to equations that would otherwise have no solutions with just real numbers. For instance, the equation \( x^2 + 1 = 0 \) has no real solutions, but has solutions in the complex number system: \( x = i \) and \( x = -i \).

Moreover, using complex numbers in exponential form becomes convenient for calculations, especially in engineering and physics, where they are used to describe things like electrical circuits and waveforms.

Periodic Functions

A periodic function is one that repeats its values at regular intervals. In mathematics, functions like sine and cosine are classic examples of periodic functions. They repeat every \( 2\pi \), the full circle in radians.

This repeating nature is crucial in understanding waveforms and oscillations, making periodic functions indispensable in fields like signal processing and communications.

• The period is the length of one complete cycle.
• For sine and cosine, this cycle length is \( 2\pi \).

In the realm of complex exponentials, periodicity comes into play significantly. The expression \( e^{ix} \) manifests periodicity through its reliance on sine and cosine, which means it completes a full cycle every \( 2\pi \) radians. This inherent periodicity of complex exponentials is a pivotal concept in determining all possible values for expressions like \( e^{i\theta} \).

Exponential Functions

Exponential functions take the form \( f(x) = b^x \), where \( b \) is a positive real number. They are characterized by a constant rate of growth or decay, depending on the value of \( b \).

Their most intriguing property is that they have constant percentage growth over equal intervals. This is not only true for the typical real-number exponents but also for complex exponents.

• Real exponents lead to straightforward growth or decay.
• Complex exponents, however, introduce rotations and oscillations.

When exponential functions have complex exponents, they can be rewritten using Euler's formula. For instance, in the expression \( 3^{i/\pi} \), the base is a real number, and the exponent involves the imaginary unit \( i \). By rewriting this using the natural logarithm and Euler's formula, one can analyze and simplify the expression to understand its periodic and oscillating nature, as well as calculate its values visually or numerically.