Problem 24 - Advanced Engineering Mathematics

Question

Along the line \(x=1\),\(\lim _{z \rightarrow 1} \frac{x+y-1}{z-1}=\lim _{y \rightarrow 0} \frac{y}{i y}=\frac{1}{i}=-i\),whereas along the \(x\) -axis, \(\lim _{z \rightarrow 1} \frac{x+y-1}{z-1}=\) \(\lim _{x \rightarrow 1} \frac{x-1}{x-1}=1.\)

Step-by-Step Solution

Answer

The limits differ along different paths, indicating discontinuity as \(z \to 1\).

Step 1: Understand the Problem

We have two limits that need to be evaluated as the complex variable \(z = x + iy\) approaches 1. Each limit is analyzed along different paths in the complex plane.

Step 2: Evaluate Limit Along the Line x = 1

Along the line \(x = 1\), we have \(z = 1 + iy\). We substitute these values into the expression \(\frac{x+y-1}{z-1}\) and evaluate the limit as \(y \to 0\).\[\lim_{y\to 0} \frac{(1 + y) - 1}{iy} = \lim_{y\to 0} \frac{y}{iy} = \lim_{y\to 0} \frac{1}{i} = \frac{1}{i} = -i\]

Step 3: Evaluate Limit Along the x-axis

Along the x-axis, we have \(y = 0\) or \(z = x + 0i\). Substituting into the expression offers:\[\lim_{x\to 1} \frac{x + 0 - 1}{x + 0 - 1} = \lim_{x\to 1} \frac{x - 1}{x - 1} = \lim_{x\to 1} 1 = 1\]

Step 4: Compare the Limits

We've calculated that the limits along different paths do not match: one path results in \(-i\) and the other in \(1\). This indicates that the function is not continuous as \(z \rightarrow 1\) from different paths.

Key Concepts

Limits in the Complex Plane

When dealing with limits in the complex plane, we look into how a function behaves as a complex variable approaches a certain point. Unlike real numbers, the complex plane is two-dimensional, consisting of real and imaginary parts. Hence, limits in the complex context become more nuanced, as they can vary depending on the path taken towards the point.
To formally define the limit of a complex function, \( f(z) \), as \( z \) approaches some point \( z_0 \), we say: \[ \lim_{z \to z_0} f(z) = L \] if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that \( 0 < |z - z_0| < \delta \) implies \( |f(z) - L| < \epsilon \). The goal is to see this relationship holds for any path leading to \( z_0 \).
However, as evident in our exercise, different paths can lead to different limiting values. This uniqueness is crucial to understanding complex functions. It’s essential to test various paths to understand the true nature of a limit in complex analysis.

Path-dependence of Limits

The path-dependence of limits refers to cases where the value of a limit varies when approached along different paths. In the complex plane, path-dependence highlights that the limit of a complex function at some point isn't singular, unlike many real-valued functions.
In our exercise, two different paths were taken to reach the point \( z = 1 \):

• Moving along the line \( x = 1 \) (where \( z = 1 + iy \)) yielded a limit of \[ -i \]
• Approaching along the x-axis (where \( y = 0 \) or \( z = x + 0i \)) resulted in a limit of \[ 1 \]

Such variations in results, based on path choice, indicate path-dependence. This can also reveal discontinuities in complex functions. Recognizing this property helps to understand not only the function but the plane's geometry too. In essence, it underscores that when dealing with limits in the complex plane, one must evaluate multiple paths to genuinely grasp the limit’s behavior.

Continuity in Complex Functions

Continuity in complex functions is an extension of the concept familiar from real analysis. A function is continuous at a point \( z_0 \) if the limit as \( z \) approaches \( z_0 \) is equal to the function's value at that point: \( \lim_{z \to z_0} f(z) = f(z_0) \). When this holds true for all points in a region, the function is continuous throughout that region.
The case we explored in the exercise highlights a situation where continuity doesn't hold. As we move towards \( z = 1 \) from different paths:

• One path yields a result of \( -i \)
• Another gives \( 1 \)

Because these limits aren’t the same along all paths, the function can't be continuous at this point. A disruption in continuity usually signifies either a singularity or some interesting feature in the function at that point.
By understanding continuity, we gain insights into the function's behavior and structure at different points within the complex plane. It helps demarcate regions where functions behave predictably from those where they might exhibit more complex phenomena.