Question

Evaluate each of the following in \(x+i y\) form, and compare with a computer solution. $$(i-1)^{i+1}$$

Short answer

The expression in polar form simplifies to: \[2^{(i+1)/2} \left[ \cos \left( \frac{3 \pi i}{4} + \frac{3 \pi}{4}\right) + i \sin \left(\frac{3 \pi i}{4} + \frac{3 \pi}{4} \right) \right] \] which can be further simplified with hyperbolic functions.

Step-by-step solution

Express the Complex Number in Trigonometric Form

First, write the base complex number \(i-1\) in polar form. To do this, find the magnitude \(r\) and the argument \( \theta \). The magnitude \( r \) is given by \[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{2} \] The argument \ ( \theta ) \ is given by \[ \theta = \tan^{-1}(\frac{1}{-1}) = \frac{3\pi}{4} \] Therefore, \( i-1 = \sqrt{2} ( \cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}))\).

Apply the Exponent in Polar Form

To raise \( \sqrt{2} ( \cos( \frac{3\pi}{ 4 }) + i \sin( \frac{3\pi}{4} )) \) to the \( (i+1) \) power, use De Moivre's Theorem, which states: \[ (\sqrt{2})^{(i+1)} \left[ \cos \left( (i+1) \cdot \frac{3\pi}{4} \right) + i \sin \left( (i+1) \cdot \frac{3\pi}{4} \right) \right] \]

Simplify the Magnitude

Calculate the new magnitude using the properties of exponents: \[ (\sqrt{2})^{i+1} = (2^{1/2})^{i+1} = 2^{(i+1)/2} \]

Simplify the Argument

Calculate the new argument: \[ (i+1) \cdot \frac{3 \pi}{ 4} = i \cdot \frac{3 \pi}{4} + 1 \cdot \frac{3 \pi}{4} = \frac{3 \pi i}{4} + \frac{3 \pi}{4} \]

Express in Real and Imaginary Form

Since the angle contains an imaginary part, the expression \( e^{i \theta} \) can be rewritten using Euler's formula for complex exponents, so we get: \[ 2^{(i+1)/2} \left[ \cos \left( \frac{3 \pi i}{4} + \frac{3 \pi}{4} \right) + i \sin \left( \frac{3 \pi i}{4} + \frac{3 \pi}{4} \right) \right] \] Simplifying this expression will involve terms that get complicated, involving hyperbolic functions due to the imaginary exponent.

Key concepts

Polar Form

Complex numbers can be represented in different ways. One is the polar form, which is very useful for multiplication and exponentiation.
The polar form represents a complex number as:
\( r \left( \cos (\theta ) + i \sin (\theta ) \right) \)
where \( r \) is the magnitude or modulus, and \( \theta \) is the argument or angle.
In the given exercise, the complex number \( i-1\) is converted into polar form with a magnitude \( r = \sqrt{2}\) and an argument \( \theta = \frac{3\text{\pi}}{4} \).
This makes it simpler to apply complex exponentiation.

De Moivre’s Theorem

De Moivre’s Theorem is essential when dealing with powers and roots of complex numbers in polar form.
It states that for any complex number \( r \left( \cos (\theta) + i \sin (\theta) \right) \), raised to a power \ n \:
\[ \left( r \left( \cos ( \theta) + i \sin ( \theta) \right) \right)^n = r^n \left( \cos ( n \theta) + i \sin ( n \theta) \right) \]
In this case, De Moivre’s Theorem is applied to compute the power \( (i-1)^{(i+1)} \) using its polar form.
This makes the computation straightforward, sidestepping the pitfalls of direct multiplication.

Euler’s Formula

Euler’s formula beautifully connects complex exponentials with trigonometric functions. It is written as:
\[ e^{ix} = \cos(x) + i \sin(x) \]
Using Euler’s formula, we can express the complex number more intuitively and calculate exponentiation easier.
More importantly, it helps convert from polar/trigonometric form back to the \( a + bi \) form.
During the exercise, the formula assists in breaking down the expression with imaginary parts.

Trigonometric Form

The trigonometric form of a complex number is: \[ r \left( \cos ( \theta ) + i \sin ( \theta ) \right) \].
This form expresses a complex number using its magnitude and angle, making it easier for certain calculations.
The exercise converts \( i-1 \) to trigonometric form before applying an exponent. Using trigonometric form and De Moivre’s theorem provides a powerful combination for handling complex-number powers.

Complex Exponentiation

Complex exponentiation extends the concept of raising numbers to powers into the complex field.
It encompasses expressions like \( (a+bi)^{(c+di)} \).
The exercise illustrates this process by raising \((i-1)\) to the power \( (i+1) \).
Rewriting the base to its polar form simplifies the exponentiation, and Euler’s Formula helps manage the imaginary components.