Problem 6 - Complex Analysis for Mathematics and Engineering

Question

Let \(z_{1}=\left(x_{1}, y_{1}\right)\) and \(z_{2}=\left(x_{2}, y_{2}\right)\) be arbitrary complex numbers. Prove or disprove the following. (a) \(\operatorname{Re}\left(z_{1}+z_{2}\right)=\operatorname{Re}\left(z_{1}\right)+\operatorname{Re}\left(z_{2}\right)\) (b) \(\operatorname{Re}\left(z_{1} z_{2}\right)=\operatorname{Re}\left(z_{1}\right) \operatorname{Re}\left(z_{2}\right)\). (c) \(\operatorname{lm}\left(z_{1}+z_{2}\right)=\operatorname{lm}\left(z_{1}\right)+\operatorname{lm}\left(z_{2}\right)\). (d) \(\operatorname{Im}\left(z_{1} z_{2}\right)=\operatorname{Im}\left(z_{1}\right) \operatorname{Im}\left(z_{2}\right)\).

Step-by-Step Solution

Answer

Statements (a) and (c) are true; (b) and (d) are false.

Step 1: Analyze Part (a)

We are given two complex numbers \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\). Their sum is \(z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)\). The real part of \(z_1 + z_2\) is \(x_1 + x_2\). Analyzing the real parts separately, \(\operatorname{Re}(z_1) = x_1\) and \(\operatorname{Re}(z_2) = x_2\). Thus, \(\operatorname{Re}(z_1 + z_2) = \operatorname{Re}(z_1) + \operatorname{Re}(z_2)\). This statement is true.

Step 2: Analyze Part (b)

For the product of two complex numbers \(z_1 \cdot z_2 = (x_1 + iy_1)(x_2 + iy_2) = x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)\), the real part is \(x_1x_2 - y_1y_2\). On the other hand, \(\operatorname{Re}(z_1) \cdot \operatorname{Re}(z_2) = x_1 \cdot x_2\). Since \(x_1x_2 - y_1y_2 eq x_1x_2\) in general, the statement \(\operatorname{Re}(z_1z_2) = \operatorname{Re}(z_1)\operatorname{Re}(z_2)\) is false.

Step 3: Analyze Part (c)

Here, \(\operatorname{lm}\) seems to be a typo and should be \(\operatorname{Im}\), referring to the imaginary part. For \(z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)\), the imaginary part is \(y_1 + y_2\). Separately, \(\operatorname{Im}(z_1) = y_1\) and \(\operatorname{Im}(z_2) = y_2\). Therefore, \(\operatorname{Im}(z_1 + z_2) = \operatorname{Im}(z_1) + \operatorname{Im}(z_2)\). The corrected statement is true.

Step 4: Analyze Part (d)

For the product of the complex numbers, \(z_1 \cdot z_2 = (x_1 + iy_1)(x_2 + iy_2) = x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)\), the imaginary part is \(x_1y_2 + x_2y_1\). Separately, \(\operatorname{Im}(z_1) \cdot \operatorname{Im}(z_2) = y_1 \cdot y_2\). As \(x_1y_2 + x_2y_1 eq y_1y_2\) in general, the statement \(\operatorname{Im}(z_1z_2) = \operatorname{Im}(z_1) \cdot \operatorname{Im}(z_2)\) is false.

Key Concepts

Real Part of Complex Numbers

The real part of a complex number is the component that does not involve the imaginary unit \(i\), which represents the square root of \(-1\).
To understand the real part, consider a complex number represented as \(z = x + iy\), where \(x\) is the real part and \(y\) is the imaginary part.
The real part, often denoted as \(\operatorname{Re}(z)\), is simply the value of \(x\).

• Example: For \(z = 3 + 4i\), the real part \(\operatorname{Re}(z) = 3\).

When adding two complex numbers \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\), the real parts add up to form the real part of the sum: \(\operatorname{Re}(z_1 + z_2) = x_1 + x_2\).
This reflects the rule that the real part of a sum is the sum of the real parts. However, when multiplying complex numbers, the real part of the product, \(\operatorname{Re}(z_1z_2) = x_1x_2 - y_1y_2\), is typically not just the product of the individual real parts.
By examining these operations, we can see how real parts interact differently under addition compared to multiplication in complex numbers.

Imaginary Part of Complex Numbers

The imaginary part of a complex number involves the coefficient of the imaginary unit \(i\).
In a complex number expressed as \(z = x + iy\), the term \(y\) is the imaginary part, and it is denoted by \(\operatorname{Im}(z)\).

• Example: For \(z = 3 + 4i\), the imaginary part \(\operatorname{Im}(z) = 4\).

When you add complex numbers, the imaginary parts themselves add: for \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\), \(\operatorname{Im}(z_1 + z_2) = y_1 + y_2\).
This matches with the property where the imaginary part of a sum equals the sum of the imaginary parts.
However, when dealing with multiplication, the imaginary part of a product, \(\operatorname{Im}(z_1z_2) = x_1y_2 + x_2y_1\), does not simply equate to the product of the individual imaginary parts.
This discrepancy underlines the complexity of interactions among imaginary components during multiplication.

Arithmetic Operations on Complex Numbers

Complex numbers are manipulated using similar arithmetic operations as real numbers with a few additional rules for handling the imaginary parts.
Two key operations often used are addition and multiplication. When adding two complex numbers, such as \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\), the formula for addition is straightforward.

• The real parts and the imaginary parts are separately added to give the sum: \(z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2)\).

This preserves the structure of complex numbers, ensuring both parts are combined accurately without alteration to the identity of the complex term.Multiplication, however, involves a more nuanced approach.

• When multiplying, the distributive property is applied: \(z_1 \cdot z_2 = (x_1 + iy_1)(x_2 + iy_2)\).

Here, you expand the expression similar to algebraic binomials: \[(x_1 + iy_1)(x_2 + iy_2) = x_1x_2 + x_1(iy_2) + (iy_1)x_2 - y_1y_2\].
Notice the subtraction of \(y_1y_2\); this results because \(i^2 = -1\).
One important result of this multiplication is combining terms to ensure the correct distribution of both real and imaginary components after multiplication:
\(z_1z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)\).
This expression highlights how each part is correctly calculated and combined, exemplifying the fundamental rules governing arithmetic operations on complex numbers.