Problem 26 - Advanced Engineering Mathematics

Question

Find a complex mapping from the given region \(R\) in the \(z\) -plane to the image region \(R^{\prime}\) in the \(w\) -plane. Strip \(0 \leq y \leq 4\) to the upper half-plane \(v \geq 0\)

Step-by-Step Solution

Answer

The function \(w = e^{\frac{\pi z}{4}}\) maps the strip to the upper half-plane.

Step 1: Understand the Problem

We need to find a complex function that maps the given strip in the z-plane, defined by the region \(0 \leq y \leq 4\), onto the upper half-plane in the w-plane, where \(v \geq 0\). This requires identifying transformations that achieve the desired mapping.

Step 2: Choose a Suitable Transformation

The mapping \(w = e^{\frac{\pi z}{4}}\) is a common choice for transforming a strip into the upper half-plane. This transformation is particularly effective in mapping vertical strips into a half-plane.

Step 3: Analyze the Mapping Effect

Let \(z = x + iy\). Then, \(w = e^{\frac{\pi (x + iy)}{4}} = e^{\frac{\pi x}{4}} e^{i\frac{\pi y}{4}}\). For \(0 \leq y \leq 4\), \(\frac{\pi y}{4}\) ranges from \(0\) to \(\pi\), thus rotating the point from \(0\) to \(\pi\), mapping it to the upper half in the w-plane.

Step 4: Confirm the Mapping

For \(y = 0\), \(e^{i\frac{\pi y}{4}} = e^{i \cdot 0} = 1\), and for \(y = 4\), \(e^{i\frac{\pi y}{4}} = e^{i\pi} = -1\). Hence, points in the z-plane strip are correctly mapped between the positive real axis and the negative real axis in the w-plane, confirming that the strip is mapped to the upper half-plane \(v \geq 0\).

Key Concepts

Complex Mapping

Complex mapping is a way to transform one region of the complex plane into another region using a function. In this context, you have a given strip in the complex plane, defined by the inequality for the imaginary part, such as the strip between lines where the imaginary part of a complex number (representing a vertical strip) is bounded by two values, here from 0 to 4.
This strip needs to be mapped onto another region, known as the image region, like the entire upper half-plane. Complex mappings are useful in many branches of mathematics, especially when solving problems involving complex functions and transformations. They allow us to move between different regions of the complex plane, making it easier to analyze and understand complex functions and their properties.

Transformations

Transformations in complex analysis involve finding a function that remaps the points from one region in the complex plane to another. This is done through specific functions that change or 'transform' the input complex numbers into new complex numbers.In the problem mentioned, the transformation used is the exponential function:- The exponential form \( w = e^{\frac{\pi z}{4}} \) helps in converting the given strip in the complex plane into an upper half-plane.- Such transformations are chosen based on their ability to alter the geometry of the input region as required.Key reasons to use transformations include:

• To simplify complex structures.
• To conform to desired shapes or boundaries.
• To facilitate easier analysis of complex functions.

Upper Half-Plane Mapping

The upper half-plane mapping turns a specific region of the complex plane into a simplified or transitioned format. By mapping to the upper half-plane in the \( w \)-plane, the points from the initial z-plane strip are transformed using a suitable function.
The transformation \( w = e^{\frac{\pi z}{4}} \) modifies the initial region by rotating and scaling it to lie above the real axis in the \( w \)-plane.The specific steps in confirming this mapping:

• By substituting the boundary points of the z-plane region into the transformation function, we analyze the outcomes.
• For instance, the boundaries \( y = 0 \) and \( y = 4 \) of the original strip are transformed into the real axis endpoints, confirming that the final region is indeed the upper half-plane.

Such mappings are fundamental in complex analysis as they simplify the study and computation of contours, intervals, and complex functions.

Complex Function

A complex function is a rule that assigns each input complex number exactly one output complex number. In the context of our problem, the complex function used is of exponential nature, \( w = e^{\frac{\pi z}{4}} \).Complex functions have the form \( f(z) = u(x, y) + iv(x, y) \) where \( z = x + iy \), and they can be analyzed for properties like continuity, differentiability, and more.Characteristics of complex functions include:

• Ability to transform regions dramatically.
• Facilitate operations like differentiation and integration over complex domains.
• Provide insights into the behavior of complex phenomena through conformal mappings.

This exercise highlights the power and capability of complex functions to transform and map between distinctly different geometric regions via succinct expressions.