Problem 7 - Complex Analysis

Question

We search for solutions \(x+i y, x, y \in \mathbb{R} .\) The substitution into the equation gives \(c=a+\mathrm{i} b=z^{2}=(x+\mathrm{i} y)^{2}\), i.e. the two equations \(x^{2}-y^{2}=a\) and \(2 x y=b\), in two unknowns \(x\) and \(y\). Together with \(x^{2}+y^{2}=|c|\) we get \(2 x^{2}=|c|+a\) and \(2 y^{2}=|c|-a .\) This determines \(x\) and \(y\) up to sign. There are thus in principle 4 possibilities, the two correct ones are singled out by the condition \(2 x y=b\) i.e. \(x y\) has the same sign as \(b .\) The solutions are $$ z=\pm\left(\sqrt{\frac{1}{2}(|c|+a)}+i \varepsilon \sqrt{\frac{1}{2}(|c|-a)}\right), \quad \varepsilon=\left\\{\begin{aligned} 1 & \text { if } b \geq 0 \\ -1 & \text { if } b<0 \end{aligned}\right. $$ One solves the quadratic equation \(z^{2}+\alpha z+\beta=0\) by the Babylonian identity $$ z^{2}+\alpha z+\beta=\left(z+\frac{\alpha}{2}\right)^{2}+\frac{4 \beta-\alpha^{2}}{4} $$

Step-by-Step Solution

Answer

The solutions for \( z \) are \( \pm\left(\sqrt{\frac{1}{2}(|c|+a)} + i \varepsilon\sqrt{\frac{1}{2}(|c|-a)}\right) \) where \( \varepsilon = 1 \) if \( b \geq 0 \) and \( \varepsilon = -1 \) if \( b < 0 \).

Step 1: Identify Given Complex Number

Start by considering that you have a complex number given as \( c = a + i b \). This is equivalent to \( c = z^2 = (x + i y)^2 \). The goal is to determine the real numbers \( x \) and \( y \) from this expression.

Step 2: Equations Formation

From the given equation \( c = (x + i y)^2 \), derive the equations \( x^2 - y^2 = a \) and \( 2xy = b \). This offers two simultaneous equations involving real variables \( x \) and \( y \).

Step 3: Incorporate Magnitude of Complex Number

Introduce the magnitude of the complex number \( |c| \). We form the equation \( x^2 + y^2 = |c| \), which is a key step in isolating \( x \) and \( y \).

Step 4: Solve for \( x^2 \) and \( y^2 \)

From the equations: \( x^2 + y^2 = |c| \) and \( x^2 - y^2 = a \), solve for \( x^2 \) and \( y^2 \) such that \( 2x^2 = |c| + a \) and \( 2y^2 = |c| - a \).

Step 5: Determine \( x \) and \( y \) Values

Solve \( x^2 = \frac{1}{2}(|c| + a) \) and \( y^2 = \frac{1}{2}(|c| - a) \) to get potential values for \( x \) and \( y \). Consider the sign condition indicated by \( 2xy = b \) to find the correct \( x \) and \( y \) pairs.

Step 6: Solution Verification

Verify solutions by checking the condition \( xy \) has the same sign as \( b \). This will determine which of the possible \( z \) values are valid.

Step 7: Employ Babylonian Identity

Use the Babylonian identity to solve the quadratic equation for \( z \), which is given by: \[ z^2 + \alpha z + \beta = \left(z + \frac{\alpha}{2}\right)^2 + \frac{4\beta - \alpha^2}{4} \] to confirm the solutions fulfill this condition.

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and are used in many fields including engineering, physics, and mathematics. They are expressed in the form \(a + i b\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit with the property \(i^2 = -1\).
Understanding complex numbers involves recognizing their operations such as addition, subtraction, multiplication, and division. Operations follow rules similar to those for real numbers but also require handling the imaginary unit.
For multiplication:

• The product \((x + i y)(u + i v)\) results in a complex number \((xu - yv) + i(xv + yu)\). This shows the interaction between the real and imaginary parts during multiplication, producing a new complex number based on the distributive property.

When solving problems or equations involving complex numbers, it is crucial to separate real and imaginary parts. This often results in systems of equations, which are solved for real and imaginary components separately. Using the modulus, or magnitude \(|c| = \sqrt{a^2 + b^2}\), can simplify these calculations, providing insights into the properties of the complex number.

Quadratic Equations

Quadratic equations in complex analysis often resemble their real-number counterparts but involve complex coefficients or solutions.
A typical form for quadratic equations is \(az^2 + bz + c = 0\). These can be solved using the quadratic formula:
\[z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The discriminant \(b^2 - 4ac\) plays a crucial role in determining the nature of the roots:

• If the discriminant is positive, the quadratic has two distinct real roots.
• If zero, exactly one repeated real root.
• If negative, the roots are complex conjugates.

In the context of our problem, we solve a quadratic equation of the form \(z^2 + \alpha z + \beta = 0\) using the Babylonian identity, which provides a clever way to refactor the equation for easier analysis of solutions using square completion and indicates whether real or complex solutions are expected.

Babylonian Identity

The Babylonian identity is a method used to simplify quadratic equations, especially handy when dealing with complex numbers. This identity transforms the quadratic equation into a more workable form by completing the square.
In practice, the identity states:
\[z^2 + \alpha z + \beta = \left(z + \frac{\alpha}{2}\right)^2 + \frac{4\beta - \alpha^2}{4}\]
This transformation reveals more information about the solutions by showing the squared term \((z + \frac{\alpha}{2})^2\) and an adjustment term \(\frac{4\beta - \alpha^2}{4}\). If the adjustment term is positive or zero, you can find real solutions. If negative, solutions are complex.

• The identity thus helps in visualizing whether the solutions are reals or complex without directly computing them initially.

Employing this method aids in solving and understanding the nature of solutions in quadratic equations in complex form, ensuring that consistent and correct results are obtained even for more intricate scenarios involving imaginary parts.