Problem 13 - Complex Analysis for Mathematics and Engineering

Question

Show that if \(z \neq 0\), then (a) \(\operatorname{Arg}(z \bar{z})=0\). (b) \(\operatorname{Arg}(z+\bar{z})=0\) when \(\operatorname{Re}(z)>0\).

Step-by-Step Solution

Answer

(a) \(\operatorname{Arg}(z \bar{z})=0\). (b) \(\operatorname{Arg}(z+\bar{z})=0\) if \(\operatorname{Re}(z)>0\).

Step 1: Understand the argument function, Arg

The argument of a complex number is the angle the line representing the complex number makes with the positive real axis in the complex plane. For a complex number \( z = x + yi \), the principal value of the argument, \( \operatorname{Arg}(z) \), lies in the range \(-\pi \leq \operatorname{Arg}(z) < \pi \).

Step 2: Prove part (a): \(\operatorname{Arg}(z \bar{z})=0\)

Given that \( z = x + yi \) and \( \bar{z} = x - yi \), the product \( z \bar{z} = (x+yi)(x-yi) = x^2 + y^2 \). This is a real number, which lies on the positive real axis. Therefore, the angle \( \theta \) from the positive real axis is zero. Hence, \( \operatorname{Arg}(z \bar{z}) = 0 \).

Step 3: Prove part (b) by examining real part condition

Consider \( z = x + yi \). Hence, \( \bar{z} = x - yi \). Adding these gives \( z+\bar{z} = (x+yi)+(x-yi) = 2x \). This is a real number equal to \( 2x \), lying on the real axis. Since \( \operatorname{Re}(z) > 0 \), \( x > 0 \) so \( z+\bar{z} > 0 \). Thus, the argument is 0. Therefore, \( \operatorname{Arg}(z+\bar{z}) = 0 \).

Key Concepts

Argument Function

The argument function, often denoted as \( \operatorname{Arg}(z) \), is a way of describing the direction of a complex number when represented in the complex plane. Imagine each complex number as a vector from the origin of a two-dimensional space. The argument tells us the angle (\( \theta \)) this vector makes with the positive x-axis (real axis) of the complex plane.

• The argument helps us understand the complex number's orientation.
• It is measured in radians, covering a principal range from \(-\pi\) to just below \(\pi\).

In the complex number \( z = a + bi \), \( a \) is the real part and \( bi \) is the imaginary part. For example, if you had a number like \( z = 3 + 4i \), its argument is found by considering the angle in the coordinate system. A special property of complex numbers is that when you multiply \( z \) by its conjugate \( \bar{z} \), the result is a real number, leading \( \operatorname{Arg}(zz) \) or \( \operatorname{Arg}(z\bar{z}) \) to be zero, as it lines up directly along the real axis. This is what was shown in part (a) of the exercise.

Complex Conjugate

The complex conjugate of a complex number \( z = a + bi \) is the number \( \bar{z} = a - bi \). It is another complex number with the real part the same but the imaginary part having the opposite sign.

• For any complex number, the product with its conjugate is always a positive real number, known as the modulus squared of the complex number.
• This property is expressed as \( z \bar{z} = a^2 + b^2 \).

This conjugate property is essential for simplifying expressions and solving problems involving complex numbers. In the given exercise, the conjugate plays a key role in showing that multiplying a complex number by its conjugate results in a non-negative real number. Hence, the argument of such a product, \( \operatorname{Arg}(z\bar{z}) \), becomes zero, as demonstrated for part (a). The conjugate operation effectively reflects the complex number across the real axis, which aids in various mathematical applications.

Real Part of Complex Number

The real part of a complex number refers to the \( a \) in the expression \( z = a + bi \). This part represents the horizontal component in the complex plane.

• To identify whether a complex expression results in a real number, you sum it with its conjugate, resulting in \( 2a \).
• If \( a > 0 \), the sum \( 2a \) is positive, lying along the positive real axis.

The exercise showed that when the real part is positive, adding a complex number and its conjugate gives a positive result on the real number line. This makes the argument of their sum, \( \operatorname{Arg}(z+\bar{z}) \), equal to zero. In essence, the real part influences the overall position of a complex number on the real axis, making it crucial for determining the argument when operations result in real values, especially in exercises like part (b).