Problem 39 - Advanced Engineering Mathematics

Question

In Problems 39-42, find all values of the given quantity. \((-i)^{4 i}\)

Step-by-Step Solution

Answer

The value is \(e^{2\pi}\), which is a real number.

Step 1: Understand the Problem

We need to find the value of the expression \((-i)^{4i}\). This involves using complex exponential functions to simplify the expression.

Step 2: Rewrite the Base in Polar Form

The base \(-i\) can be expressed in its polar form. The complex number \(-i\) has a magnitude of 1 and an angle \(-\frac{\pi}{2}\). So, it can be written as \(e^{-i\frac{\pi}{2}}\).

Step 3: Apply the Power to the Polar Form

Now apply the exponent \(4i\) to the polar form: \[((e^{-i\frac{\pi}{2}})^{4i}).\] Recall that \((e^{a})^{b} = e^{ab}\), so the expression becomes \(e^{4i(-i\frac{\pi}{2})}\).

Step 4: Simplify the Exponent

Simplify the expression \(4i(-i\frac{\pi}{2})\). Since \(i^2 = -1\), this becomes \(-4\times -\frac{\pi}{2} = 2\pi\).

Step 5: Evaluate the Simplified Exponent

Now compute \(e^{2\pi}\). This is a real number because the complex exponential returns to real values over multiples of \(2\pi\). Thus, \(e^{2\pi}\) is a constant real value.

Key Concepts

Polar Form of Complex Numbers

Rewriting a complex number into its polar form makes it easier to handle, especially for multiplication and exponentiation.
The polar form represents a complex number using a magnitude and an angle. The magnitude of the complex number \(-i\) is \(1\) because it is one unit away from the origin on the complex plane.
The angle, or argument, of \(-i\) is \(-\frac{\pi}{2}\) because it lies directly downwards on the imaginary axis.
Thus, \(-i\) can be represented in polar form as \(|-i|e^{i(-\frac{\pi}{2})}\), or simply \(e^{-i\frac{\pi}{2}}\) when the magnitude is \(1\).
Knowing how to convert a complex number to polar form enables us to use exponential properties to simplify expressions.

Exponential Properties

When dealing with powers in exponential form, remember the essential property: \((e^{a})^{b} = e^{ab}\).
This property allows us to multiply the exponents directly rather than dealing with cumbersome multiplication of trigonometric expressions.
In the given exercise, the base of the expression was written in polar form as \(e^{-i\frac{\pi}{2}}\). By raising it to the power of \(4i\), the expression becomes \(e^{4i(-i\frac{\pi}{2})}\).
Simplifying inside the exponent is crucial. Here, multiplying \(4i \) by \(-i\) results in using \(i^2 = -1\), transforming the expression into numeric terms.

Complex Numbers Powers

Raising complex numbers to a power can be simplified through their polar representation.
In this exercise, we derived the expression \(e^{4i(-i\frac{\pi}{2})}\) using the polar form and exponential properties, resulting in \(e^{2\pi}\).
This simplification should help to visualize: the complex exponential function effectively cycles through, leading us back to real numbers when the angle is a multiple of \(2\pi\).

• The property \(i^2 = -1\) simplifies calculations.
• Euler's formula connects exponential expressions to trigonometric functions, enhancing intuition.
• The periodicity of the exponential function \((e^{i\theta})\) allows simplifications involving multiples of \(2\pi\)

Thus, understanding these properties and relationships helps make sense of complex number powers, facilitating easier computation.