Problem 7 - Complex Analysis

Question

Square roots and the solvability of quadratic equations in \(\mathbb{C}\) Let \(c=a+\mathrm{i} b \neq 0\) be a given complex number. By splitting it into its real and imaginary part show that there are exactly two complex numbers \(z_{1}\) and \(z_{2}\) such that $$ z_{1}^{2}=z_{2}^{2}=c . \text { One has } z_{2}=-z_{1} $$ \(\left(z_{1}\right.\) and \(z_{2}\) are called the square roots of \(c\) ) For example, determine the square roots of $$ 5+7 \mathrm{i}, \quad \text { and } \quad \sqrt{2}+\mathrm{i} \sqrt{2} $$ Use also polar coordinates for this exercise. Furthermore, show that a quadratic equation $$ z^{2}+\alpha z+\beta=0, \quad \alpha, \beta \in \mathbb{C} \text { arbitrary } $$ always has at most two solutions \(z_{1}, z_{2} \in \mathrm{C}\).

Step-by-Step Solution

Answer

Every complex number has two distinct square roots in \( \mathbb{C}\). Quadratic equations in \( \mathbb{C} \) have at most two solutions.

Step 1: Express the Complex Number in Polar Form

Given a complex number \( c = a + ib \), we can express it in polar form as \( c = re^{i\theta} \) where \( r = \sqrt{a^2 + b^2} \) is the magnitude of \( c \), and \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \) is the argument.

Step 2: Determine the Square Roots in Polar Form

To find the square roots of \( c \), we set \( z = \sqrt{r} e^{i\frac{\theta}{2}} \). The square roots are given by \( z_1 = \sqrt{r} e^{i\frac{\theta}{2}} \) and \( z_2 = \sqrt{r} e^{i\left(\frac{\theta}{2} + \pi\right)} = -z_1 \).

Step 3: Verify the Calculation for Square Roots

For \( 5 + 7i \), first determine \( r = \sqrt{5^2 + 7^2} = \sqrt{74} \) and \( \theta = \tan^{-1}\left(\frac{7}{5}\right) \). Use these in the formula \( z_1 = \sqrt{\sqrt{74}} e^{i\frac{\theta}{2}} \) and \( z_2 = -z_1 \). Repeat similarly for \( \sqrt{2} + i\sqrt{2} \).

Step 4: Solve the Quadratic Equation Using the Quadratic Formula

For the quadratic equation \( z^2 + \alpha z + \beta = 0 \), use the quadratic formula \( z = \frac{-\alpha \pm \sqrt{\alpha^2 - 4\beta}}{2} \). This demonstrates that there can be zero, one, or two solutions depending on the value of the discriminant (\( \alpha^2 - 4\beta \)).

Step 5: Conclude with the Number of Solutions

By the nature of the quadratic formula, a quadratic equation in \( \mathbb{C} \) has at most two distinct solutions, corresponding to the square roots of the constant term in polar representation as shown.

Key Concepts

Quadratic Equations

Quadratic equations hold a special place in mathematics, including when dealing with complex numbers. Quadratic equations have the standard form given by \[ z^2 + \alpha z + \beta = 0 \] where \( \alpha \) and \( \beta \) are complex numbers. A notable feature of quadratic equations is that they have at most two solutions.

These solutions are found using the well-known quadratic formula:\[z = \frac{-\alpha \pm \sqrt{\alpha^2 - 4\beta}}{2}\] This formula works in the realm of complex numbers just as it does with real numbers. How many solutions there are depends on the discriminant, \( \alpha^2 - 4\beta \):

• If the discriminant is zero, there's one unique solution.
• If the discriminant is positive, there are two distinct solutions.
• If the discriminant is negative, there are two distinct complex solutions which are complex conjugates of each other.

In any case, the maximum number of solutions remains two. Understanding how to handle the discriminant when it's a complex number is crucial for effectively solving quadratic equations in the complex plane.

Polar Coordinates

Polar coordinates offer a powerful way to understand and manipulate complex numbers. Any complex number \( c = a + ib \) can be converted into polar form. This is represented as \[ c = re^{i\theta} \] where \( r = \sqrt{a^2 + b^2} \) is the magnitude (or modulus) of the complex number, and \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \) is the argument or angle.

Using polar coordinates, complex multiplication and division become much simpler, involving just the addition or subtraction of angles. This ease extends to finding roots, such as square roots: by halving the angle for square roots in the polar form. For example, to find the square root of \( c \), you find \( z = \sqrt{r} e^{i\frac{\theta}{2}} \).

Polar coordinates bring clarity especially when visualizing complex numbers on the complex plane. Here each complex number is represented as a point or vector originating from the origin, making operations like rotation (changing the angle) intuitive.

Square Roots of Complex Numbers

Finding the square roots of complex numbers can be tackled efficiently using polar coordinates. Any complex number \( c = a + ib \) has exactly two square roots \( z_1 \) and \( z_2 \), such that \[ z_1^2 = z_2^2 = c \]where \( z_2 = -z_1 \).

The process to find these roots involves:

• First, express the complex number in its polar form as \( c = re^{i\theta} \).
• Then, compute the square roots using the formula \( z = \sqrt{r} e^{i\frac{\theta}{2}} \).
• The second root is given by \( z_2 = \sqrt{r} e^{i\left(\frac{\theta}{2} + \pi\right)} \), which equals \(-z_1\).

This ensures each complex number has two distinct roots that are negatives of each other, reflecting symmetry in the complex plane.