Problem 6 - Complex Analysis for Mathematics and Engineering

Question

Show that arg \(z_{1}=\arg z_{2}\) iff \(\mathrm{z}_{2}=c z_{1}\), where \(c\) is a positive real constant.

Step-by-Step Solution

Answer

\(\arg(z_1) = \arg(z_2)\) if and only if \(z_2 = cz_1\), where \(c\) is positive.

Step 1: Understanding the Argument of a Complex Number

For any complex number \(z = x + yi\), the argument, \(\arg z\), is the angle \(\theta\) such that \(\tan(\theta) = \frac{y}{x}\). This angle can also be obtained using \(\theta = \text{atan2}(y, x)\), which accurately gives the angle in the correct quadrant.

Step 2: Define the Equality of Arguments

The exercise states that \(\arg(z_1) = \arg(z_2)\). We have \(z_1 = x_1 + y_1i\) and \(z_2 = x_2 + y_2i\). This means that both complex numbers have the same angle with the positive real axis.

Step 3: Relate \(z_2\) to \(z_1\) with a Positive Constant

If \(\arg(z_1) = \arg(z_2)\), it implies that \(z_1\) and \(z_2\) lie on the same line in the complex plane passing through the origin. Hence, \(z_2\) can be expressed as \(z_2 = cz_1\) where \(c\) is a positive real constant. This is because only a scaling factor that is positive (keeping the angle unchanged) can map one vector to another along the same line.

Step 4: Proof of the "if and only if" Condition

For the reverse direction, if \(z_2 = cz_1\) where \(c > 0\), then the magnitude of \(z_2\) is scaled by \(c\) but the argument remains the same since multiplying by a positive real number does not change the angle. Hence, \(\arg(z_1) = \arg(z_2)\) still holds. Therefore, we have shown that "if and only if" condition holds. $$

Key Concepts

Argument of a Complex Number

Understanding the argument of a complex number is important when exploring how complex numbers interact geometrically. Each complex number can be expressed as \(z = x + yi\), where \(x\) and \(y\) are real numbers and \(i\) is the imaginary unit satisfying \(i^2 = -1\). The argument \(\arg z\) is essentially the angle \(\theta\) between the positive real axis and the line joining the origin to the point \((x, y)\) on the complex plane. This can be calculated using the arctangent function: \(\theta = \arctan\left(\frac{y}{x}\right)\).

• Remember that \(\arg z\) is typically measured in radians.
• Functions like \(\text{atan2}(y, x)\) help identify the angle in the correct quadrant.

The argument plays a crucial role as it determines the direction of the complex number in the plane, providing insight into its geometric interpretation.

Complex Plane

The complex plane is a graphical representation where complex numbers are visualized as points. The plane itself has two axes, with the horizontal axis representing the real part and the vertical axis representing the imaginary part of a complex number.

• Each point or vector \((x, y)\) on this plane corresponds to the complex number \(z = x + yi\).
• The distance of a point from the origin, known as the modulus \(|z|\), is given by \(|z| = \sqrt{x^2 + y^2}\).
• Complex addition, subtraction, and multiplication can be visualized via basic geometric operations on this plane.

Using the complex plane helps with visualizing the effects of operations on complex numbers, like rotations and scaling, making abstract operations more intuitive.

Scaling in Complex Numbers

Scaling is an operation that involves changing a complex number's size while maintaining its direction. In the context of complex numbers, scaling involves multiplying the number by a real constant. If you have a complex number \(z_1 = x_1 + y_1i\) and you want to obtain another complex number \(z_2\) with the same argument, you would express \(z_2\) as \(z_2 = cz_1\), where \(c\) is a positive real number.

• This operation does not alter the argument because multiplying by a positive real number scales the magnitude but not the angle.
• Graphically, both points \(z_1\) and \(z_2\) will lie on the same line originating from the origin, in the complex plane.

The concept of scaling shows how multiplication by a positive scalar changes both size and the location of a complex number on the complex plane without changing its orientation.