Problem 11 - Advanced Engineering Mathematics

Question

\(\int_{C} e^{2} d z\), where \(C\) is the polygonal path consisting of the line segments from \(z=0\) to \(z=2\) and from \(z=2\) to \(z=1+\pi i\)

Step-by-Step Solution

Answer

The integral is \( e^2 (2 + \sqrt{1 + \pi^2}) \).

Step 1: Interpret the Problem

We need to evaluate the line integral \( \int_{C} e^2 \, dz \), where \( C \) is the path from \( z = 0 \) to \( z = 2 \) and from \( z = 2 \) to \( z = 1 + \pi i \). The function being integrated, \( e^2 \), is a constant with respect to \( z \).

Step 2: Understand Line Integral of a Constant

Since \( e^2 \) is constant, the integral becomes \( e^2 \int_C 1 \, dz \) which is equal to \( e^2 \times \text{(length of } C\text{)} \). This reduces the problem to finding the length of the path \( C \).

Step 3: Calculate Length of Path (Segment 1)

For the first segment from \( z = 0 \) to \( z = 2 \), the length is simply the distance between these two points in the complex plane, which is \(|2 - 0| = 2\).

Step 4: Calculate Length of Path (Segment 2)

For the second segment from \( z = 2 \) to \( z = 1 + \pi i \), the length is \(|1 + \pi i - 2| = \sqrt{(1-2)^2 + (\pi - 0)^2} = \sqrt{1 + \pi^2}\).

Step 5: Sum Distances to Find Total Path Length

Add the lengths of the two segments to find the total length of the path \( C \):\[2 + \sqrt{1 + \pi^2}\]

Step 6: Evaluate the Integral

Substitute the total length back into the integral expression: \[\int_C e^2 \, dz = e^2 \times (2 + \sqrt{1 + \pi^2})\]

Key Concepts

Polygonal Path

A polygonal path in the context of complex analysis refers to a piecewise linear curve consisting of multiple line segments connected end to end. It's like connecting the dots with straight lines in the complex plane to create a path with a specific shape.
In our given exercise, the path \( C \) is a polygonal path composed of two segments:

• The line from \( z = 0 \) to \( z = 2 \).
• The line from \( z = 2 \) to \( z = 1 + \pi i \).

By breaking down a path into linear segments, we can simplify calculations, especially when dealing with line integrals. Each segment can be analyzed separately, making it easier to work out the overall properties of the path.

Complex Plane

The complex plane is a two-dimensional plane where every point represents a complex number. The horizontal axis or x-axis holds the real part of the complex number, and the vertical axis or y-axis represents the imaginary part.
This way of visualizing complex numbers allows us to use geometric interpretations for complex operations.
Key features of the complex plane include:

• Each point \( z \) can be described as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part.
• Distance and angles can be measured directly, making it a powerful tool for solving problems involving complex numbers.

In our example, the path includes points like \( z = 0 \), \( z = 2 \), and \( z = 1 + \pi i \), plotted using this plane to help calculate distances and integrals.

Constant Function

A constant function is a function that yields the same value no matter what the input is. In our case, the function being integrated is \( e^2 \), which is a constant because it does not depend on the variable \( z \).
When dealing with such functions in integral calculus, specifically in line integrals, some key properties help simplify calculations:

• The integral of a constant function over a path equals the value of the constant multiplied by the total length of the path.
• This property reduces computational complexity since the function's behavior doesn't change over the path.

In our exercise, the constant \( e^2 \) allows us to solely focus on determining the length of the path \( C \), rather than dealing with varying function values along the path.

Path Length Calculation

Path length calculation involves determining the total length of a specified path. For the exercise's polygonal path, this means summing the lengths of all its segments.
For each segment, the distance formula in the complex plane is used. Here’s how it works:

• For a segment between points \( z_1 = x_1 + y_1 i \) and \( z_2 = x_2 + y_2 i \), the length is \( |z_2 - z_1| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Let's break down our exercise:

• The first segment length is \(|2 - 0| = 2\) since it lies along the real axis with no imaginary component.
• The second segment from \( z = 2 \) to \( z = 1 + \pi i \) uses the same distance formula: \( \sqrt{(1-2)^2 + (\pi - 0)^2} = \sqrt{1 + \pi^2} \).

Adding these gives the total path length \( 2 + \sqrt{1 + \pi^2} \), an essential step in evaluating the given line integral.