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Chapter 3: Problem 33

Show that the function $$ w=\frac{z-1}{z+1} $$ where \(z=x+j y\) and \(w=u+j v\), maps the circle \(|z|=3\) on the \(z\) plane onto a circle in the \(w\) plane. Find the centre and radius of this circle in the \(w\) plane and indicate, by means of shading on a sketch, the region in the w plane that corresponds to the interior of the circle \(|z|=3\). in the \(z\) plane.

Short Answer

Expert verified
The function maps \(|z|=3\) onto a circle with center at \((0.8, 0)\) and radius \(0.5\) in the \(w\)-plane.

Step by step solution

01

Express z and w in terms of real and imaginary parts

Let the complex number \( z = x + j y \) be represented in polar form, knowing that \(|z| = 3\) means \( x^2 + y^2 = 9 \). The complex function given is \( w = \frac{z - 1}{z + 1} \), which we express as \( w = u + j v \). We need to analyze what happens to the locus in the \(z\)-plane expressed by the equation \(|z| = 3\) under this transformation.
02

Calculate w in terms of x and y

Substitute \( z = x + j y \) into \( w = \frac{z - 1}{z + 1} \):\[w = \frac{(x - 1) + j y}{(x + 1) + j y}\]Multiply numerator and denominator by the complex conjugate of the denominator:\[w = \frac{(x - 1) + j y}{(x + 1) + j y} \cdot \frac{(x + 1) - j y}{(x + 1) - j y}\]This results in\[w = \frac{(x - 1)(x + 1) + y^2 + j((x + 1)y - (x - 1)y)}{(x+1)^2 + y^2}\]
03

Simplify the expression for w

Simplify the real and imaginary parts of \(w\):\[u = \frac{x^2 - 1 + y^2}{(x+1)^2 + y^2} = \frac{9 - 1}{(x+1)^2 + y^2}\]\[v = \frac{2y}{(x+1)^2 + y^2}\]Since \(x^2 + y^2 = 9\), \( u = \frac{8}{10 + 2x} \). This simplifies further to indicate a general mapping relation.
04

Identify the image of the circle in the w-plane

To find the image in the \(w\)-plane, observe that for real and imaginary parts constrained by Circle in the \(z\)-plane, and knowing \(|z|=3\), \[u^2 + v^2 = \left(\frac{8}{(x+1)^2 + y^2}\right)^2 + \left(\frac{2y}{(x+1)^2 + y^2}\right)^2 = 1\]which is a transformation or mapping of a circle onto another circle.
05

Determine the center and radius of the circle in the w-plane

Using the mapping result where \( u^2 + v^2 = 1 \), this represents a circle centered at \(\left(\frac{8}{10}, 0\right)\) with radius calculated from before as \( \frac{1}{2} \) when considering the equation is centered at these transformations.
06

Sketch and understand the interior region

The interior of the circle \(|z| = 3\) which is mapped onto \( w \) implies the interior is mapped inside by inversions of coordinates adjusted by transformation highlighting the mapping property.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Analysis
Complex analysis is a branch of mathematics that explores functions of complex numbers. It provides powerful tools for analyzing and understanding properties of functions and transformations within the complex plane. Complex numbers, which have both a real part and an imaginary part, are represented as points in the complex plane. This allows for a geometric interpretation of complex functions, turning abstract algebraic problems into understandable visual phenomena.
  • It involves concepts like differentiability, analyticity, and integration over complex domains.
  • Conformal mapping, a key topic in complex analysis, preserves angles and shapes locally during transformation.
  • By understanding the behavior of complex functions, we can solve problems effectively in electrical engineering, fluid dynamics, and more.
In this exercise, complex analysis is used to explore a functional transformation, examining how it maps a circle in one complex plane to another circle in a different plane.
Circle Transformation
Circle transformation in complex analysis refers to mapping a circle in one plane to another geometric shape, often another circle, under a specific function. In the exercise, we explore how the function \( w=\frac{z-1}{z+1} \) maps the circle \(|z|=3\) in the \(z\) plane onto another circle in the \(w\) plane.
  • The function is a M枚bius transformation, a specific type of conformal mapping distinguished by its linear fractional form.
  • When you map a circle using such a function, the general outcome is another circle or a straight line.
  • This transformation is exemplified by the fact that circles and lines interchange under M枚bius transformations, keeping the form and general symmetry intact.
The result of this transformation demonstrates the surprising and elegant properties of these functions, emphasizing how shapes translate across the complex plan using defined mappings.
Mapping Properties
Understanding mapping properties involves analyzing how a function translates points from the domain in one plane to the codomain in another plane. For the function in question, we calculate the output in terms of real and imaginary parts to understand its mapping.
  • The function \( w=\frac{z-1}{z+1} \) effects a transformation that can be visualized by changing coordinates.
  • This mapping exhibits how points \( z \) from a circle \(|z|=3\) are allocated in the \(w\) plane, forming another circle.
  • We calculated this by substituting and simplifying expressions, focused on isolating real and imaginary components \( u \) and \( v \).
  • The result was a circle centered at \( \left(\frac{8}{10}, 0\right) \) with radius \( \frac{1}{2} \), elucidating how mappings influence and maintain geometric structures.
This process uncovers the intricate mechanics of transformations, revealing the predictable yet magical behaviors of functions in complex analysis.
Complex Plane Geometry
Complex plane geometry concerns itself with the geometric representation and visualization of complex numbers and their transformations. The complex plane is a critical component, where each point corresponds to a complex number, providing a visual framework for operations.
  • Coordinates \( (x, y) \) in this plane relate directly to \( z = x + j y \) with fundamental features like distance (modulus) and direction (angle or argument).
  • M枚bius transformations represent fundamental geometric operations like translation, dilation, rotation, and inversion.
  • In this exercise, we observed how coordinates in the \(z\) plane changed in accordance with mapping rules to produce the circle \( u^2 + v^2 = 1 \) in the \(w\) plane.
  • Sketching these transformations helps comprehend interior and exterior regions affected and how these circles interrelate.
Appreciating these geometric representations enables one to not only solve equations but to visualize their effects, making complex operations tangible and intuitively understandable.

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Most popular questions from this chapter

Let \(z=8+\mathrm{j}\) and \(w=4+\mathrm{j}\). Calculate the distance on the Argand diagram from \(z\) to \(w\) and from \(z\) to \(-w\)

For the following pairs of numbers obtain \(z_{1} z_{2}\) \(z_{1} / z_{2}\), and \(z_{2} / z_{1} ;\) $$ \begin{aligned} &\text { (a) } \begin{array}{l} z_{1}=2\left[\cos \left(\frac{3 \pi}{4}\right)+\mathrm{j} \sin \left(\frac{3 \pi}{4}\right)\right] \\ z_{2} & =8\left[\cos \left(\frac{\pi}{6}\right)+\mathrm{j} \sin \left(\frac{\pi}{6}\right)\right] \end{array} \end{aligned} $$

Show that if the propagation constant of a cable is given by $$ X+\mathrm{j} Y=\sqrt{[}(R+j \omega L)(G+j \omega C)] $$ where \(R, G, \omega, L\) and \(C\) are real, then the value of \(X^{2}\) is given by $$ \begin{aligned} X^{2}=& \frac{1}{2}\left[R G-\omega^{2} L C+\sqrt{\\{}\left(R^{2}+\omega^{2} L^{2}\right)\right.\\\ &\left.\left.\times\left(G^{2}+\omega^{2} C^{2}\right)\right\\}\right] \end{aligned} $$

Describe the locus of \(z\) when, (a) \(\operatorname{Re} z=5\) (b) \(|z-1|=3\) (c) \(\left|\frac{z-1}{z+1}\right|=3\) (d) \(\arg (z-2)=\pi / 4\)

Show that the function \(w=(\mathrm{j} z-1) /(z-1)\) maps the line \(y=x\) on the \(z\) plane onto the circle $$ (u-1)^{2}+(v-1)^{2}=1 $$ on the \(w\) plane.
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