Problem 31 - Advanced Engineering Mathematics

Question

\(f(z)=x+\frac{x}{x^{2}+y^{2}}+i\left(y-\frac{y}{x^{2}+y^{2}}\right) .\) The level curve \(v(x, y)=0\) is described by \(y-\frac{y}{x^{2}+y^{2}}=0\) or \(y\left(x^{2}+y^{2}-1\right)=0 . \quad\) We see that either \(y=0\) or \(x^{2}+y^{2}=1 .\) Thus \(v(x, y)=0\) gives either the \(x\) -axis (without the origin (0,0) ) or the unit circle \(x^{2}+y^{2}=1\)

Step-by-Step Solution

Answer

The level curves are the x-axis (excluding the origin) and the unit circle.

Step 1: Understanding the Exercise

The function given is complex, represented as \( f(z) = x + \frac{x}{x^2 + y^2} + i\left(y - \frac{y}{x^2 + y^2}\right) \). We need to examine the level curve where \( v(x, y) = 0 \). Here, the imaginary part is \( v(x, y) = y - \frac{y}{x^2 + y^2} \). Our task is to find when this equals zero.

Step 2: Setting the Equation to Zero

We set the equation \( y - \frac{y}{x^2 + y^2} = 0 \). Simplifying this, we multiply through by \( x^2 + y^2 \) to clear the denominator: \( y(x^2 + y^2) - y = 0 \), which simplifies to \( y(x^2 + y^2 - 1) = 0 \).

Step 3: Solving for Possible Criteria

The equation \( y(x^2 + y^2 - 1) = 0 \) gives us two possibilities: \( y = 0 \) or \( x^2 + y^2 = 1 \). These are the criteria for the level curve described by the imaginary part equalling zero.

Step 4: Interpreting the Solutions

The possible solutions are \( y = 0 \), which corresponds to the entire x-axis, and \( x^2 + y^2 = 1 \), the equation of a unit circle. However, the condition excludes the origin (0,0) because division by zero is undefined at the point (0,0) in the original function.

Step 5: Conclusion on Level Curves

The level curve \( v(x, y) = 0 \) consists of the x-axis (minus the origin) and the unit circle centered at the origin. Therefore, these are the two distinct geometrical interpretations of the level curves in this context.

Key Concepts

Level Curves

Level curves are a fascinating aspect of complex analysis and help us understand the behavior of complex functions. They are contours on which a function takes on a constant value.
In the context of the given exercise, the level curve refers to where the imaginary part of the complex function equals zero. This forms a visual map indicative of where this condition holds.

• For the function provided, we identify level curves by solving for conditions where the imaginary part equals zero.
• The solutions provide two geometric interpretations: the x-axis (excluding the origin due to undefined values there) and the unit circle.

Understanding level curves serves as a crucial tool in analyzing complex functions, providing insights into their structure and nature.

Imaginary Part

The imaginary part of a complex function is crucial for examining the nature of the function at different points. In our exercise, the imaginary part is represented as \( v(x, y) = y - \frac{y}{x^2 + y^2} \).
The task is to understand when this imaginary part becomes zero:

• This happens when \( y - \frac{y}{x^2 + y^2} = 0 \).
• Simplifying leads to an important condition: \( y(x^2 + y^2 - 1) = 0 \).

These conditions yield geometric shapes or curves on which the imaginary component shows a specific behavior, impacting the full representation of the complex function.

Complex Functions

A complex function is a function that involves complex numbers, typically written as \( f(z) = u(x, y) + iv(x, y) \), where \( u \) and \( v \) are real-valued functions.
For our function, \( f(z) = x + \frac{x}{x^2 + y^2} + i\left(y - \frac{y}{x^2 + y^2}\right) \), we need to focus both on the real part and the imaginary part separately:

• Real part: \( x + \frac{x}{x^2 + y^2} \).
• Imaginary part: \( y - \frac{y}{x^2 + y^2} \).

These parts give us insights into the function's behavior in the complex plane, helping describe phenomena like level curves and effects on magnitude or direction.

Unit Circle

The unit circle is a simple yet powerful concept in complex analysis. It's the set of all points that are at a distance of one from the origin in the complex plane.
In this exercise, the unit circle is represented by the equation \( x^2 + y^2 = 1 \).

• This curve is significant because it also represents where the imaginary part equals zero aside from the x-axis.
• The unit circle holds valuable properties such as symmetry and plays a vital role in various complex functions.

Understanding the role of the unit circle helps us better grasp the geometric interpretation of complex functions and their level curves.