Problem 15 - Complex Analysis for Mathematics and Engineering

Question

Find all four roots of \(z^{4}+4=0\), and use them to demonstrate that \(z^{4}+4\) can be factored into two quadratics with real coefficients.

Step-by-Step Solution

Answer

The roots are \( \pm \frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2} \). The polynomial factors as \((z^2-\sqrt{2}z+2)(z^2+\sqrt{2}z+2)\).

Step 1: Rearrange the Equation

We start with the equation \( z^4 + 4 = 0 \). Rearrange this to \( z^4 = -4 \). This indicates that we are looking for the fourth roots of \(-4\).

Step 2: Express the Complex Number in Polar Form

The complex number \(-4\) can be expressed in polar form as \(4\text{cis}(\pi)\) because its magnitude is 4 and its argument in radians is \(\pi\) (since it lies on the negative real axis).

Step 3: Use De Moivre's Theorem to Find Roots

According to De Moivre's Theorem, the nth roots of a complex number \(r\text{cis}(\theta)\) are given by \[ r^{1/n} \text{cis} \left( \frac{\theta + 2k\pi}{n} \right) \] for \(k = 0, 1, ..., n-1\). Here, \(r=4\) and \(n=4\).

Step 4: Calculate Magnitude of Roots

The magnitude of each root is \(4^{1/4} = \sqrt{2}\), as 4 is expressed as 2 squared, and the square root raised to the fourth power gives \(\sqrt{2}\).

Step 5: Calculate Each Root Using Angles

Find each root using different angles:- For \(k=0\), the angle is \(\frac{\pi}{4}\), so the root is \( \sqrt{2}\text{cis}\frac{\pi}{4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \).- For \(k=1\), the angle is \(\frac{3\pi}{4}\), so the root is \( \sqrt{2}\text{cis}\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \).- For \(k=2\), the angle is \(\frac{5\pi}{4}\), so the root is \( \sqrt{2}\text{cis}\frac{5\pi}{4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \).- For \(k=3\), the angle is \(\frac{7\pi}{4}\), so the root is \( \sqrt{2}\text{cis}\frac{7\pi}{4} = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \).

Step 6: Verify the Roots Satisfy the Original Equation

Verify that each root raised to the fourth power equals \(-4\). This is to ensure they are correct and satisfy the equation \(z^4 + 4 = 0\).

Step 7: Factor the Original Expression

Group the roots obtained in conjugate pairs to form quadratic factors with real coefficients:- Use roots \(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\) and \(-\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\) to form: \( (z - (\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}))(z - (-\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2})) = z^2 - \sqrt{2}z + 2 \).- Use roots \(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\) and \(\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\) to form: \( (z - (-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}))(z - (\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2})) = z^2 + \sqrt{2}z + 2 \).

Step 8: Verify Quadratic Factors' Back Multiplication

Multiply the quadratic factors together: \( (z^2 - \sqrt{2}z + 2)(z^2 + \sqrt{2}z + 2) \) to get back \(z^4 + 4\), ensuring the factorization is correct.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number theory. It helps us find the roots of complex numbers with ease.
Imagine having a complex number written in polar form as \( r \text{cis}(\theta) \) where \( r \) is the magnitude and \( \theta \) is the angle or argument.

According to De Moivre's Theorem, the nth roots of this complex number are calculated using the formula:

• \[ r^{1/n} \text{cis} \left( \frac{\theta + 2k\pi}{n} \right) \]

In our specific problem, we were searching for the fourth roots of \(-4\). We expressed \(-4\) in polar form as \(4\text{cis}(\pi)\), giving us a magnitude of 4 and an angle of \(\pi\).

Then we applied De Moivre's Theorem to find each root by varying \(k\) from 0 to 3:

• Calculating the magnitude of the roots as \(4^{1/4} = \sqrt{2}\)
• Determining angles by plugging into \(\frac{\pi + 2k\pi}{4}\), resulting in distinct roots.

De Moivre's Theorem not only simplifies the process of finding roots but also ensures comprehensive understanding of how these roots are derived geometrically on the complex plane.

Polar Form of Complex Numbers

The polar form is a special way of expressing complex numbers, allowing ease of calculation
in operations such as multiplication, division, and finding roots. In polar form, a complex number \( z = a + bi \) is expressed as \( r \text{cis}(\theta) \),and it relates to the Cartesian form by:

• Magnitude \( r = \sqrt{a^2 + b^2} \)
• Angle \( \theta = \tan^{-1}(\frac{b}{a}) \)

For example, the complex number \(-4\) lies on the negative real axis, making its magnitude 4 and its angle \(\pi\) or 180°.

This transformation to polar form was crucial in our exercise, as it set the stage for applying De Moivre's Theorem.

• The polar form emphasizes the symmetry and periodicity in complex roots.
• It provides clarity, especially when visualizing complex roots on the unit circle.

The polar form thus makes manipulation of complex numbers more intuitive, directly tying into how solutions like roots or power expressions are calculated.

Quadratic Factorization with Real Coefficients

The goal of factorization in our problem was to express the original quartic equation
\( z^4 + 4 = 0 \) into a product of two quadratic equations with real coefficients.
This method leverages the concept of conjugate pairs of complex roots. Complex roots appear in conjugate pairs when all coefficients in an equation are real – meaning if \( z \) is a root, \( \overline{z} \) must also be a root.

Here's how factorization occurred:

• Identify conjugate pairs: We paired \( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \) with \( -\frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \) and \( -\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \) with \( \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \).
• For each pair, form quadratic factors: \( (z - a)(z - \overline{a}) \) which expand to \( z^2 - (a + \overline{a})z + |a|^2 \), removing the complex part.

The process results in meaningful real coefficients, maintaining the integrity of the original equation, while transforming its appearance.
This step illuminated how complex numbers, even wild and imaginary, can culminate in orderly real number polynomials through thoughtful factorization and recognition of conjugate pairs.