/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Find the image of the given line... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 13

Find the image of the given line under the complex mapping \(w=z^{2}\).\(y=x\)

Short Answer

Expert verified
The image is the imaginary axis with parabolic scaling \(v = 2u^2\).

Step by step solution

01

Understand the Problem

We need to find the image of the line described by the equation \(y = x\) when it is transformed by the complex mapping \(w = z^2\). This is a transformation in the complex plane.
02

Describe the Mapping

The complex mapping \(w = z^2\) means that for every complex number \(z = x + yi\), the image \(w\) is given by \(w = (x + yi)^2 = x^2 - y^2 + 2xyi\).
03

Set the Line Equation in Complex Form

In terms of complex numbers, the line \(y = x\) can be expressed as \(z = x + xi = x(1 + i)\). This represents all points in the form \(x(y = x)\) on the complex plane.
04

Apply the Mapping to the Line

Substitute \(z = x(1 + i)\) into the mapping \(w = z^2\). We get \(w = (x(1 + i))^2 = x^2(1 + i)^2 = x^2(-2i) = -2ix^2\).
05

Simplify the Image

The expression \(-2ix^2\) can be rewritten as \(w = 2x^2i\), where the real part is zero and the imaginary part is \(2x^2\). This means the image is purely imaginary.
06

Interpret the Result

The image of the line \(y = x\) under the transformation is the imaginary axis, scaled by \(2x^2\). Hence, for a given \(x\), the point \(w\) lies along the line \(v = 2u^2\) (where \(w = u + vi\) and \(u = 0\)). This results in a parabola opening along the imaginary axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Complex Plane
The complex plane is much like a two-dimensional cartesian plane. However, it has a unique twist. Instead of traditional x and y axes, it uses the real and imaginary numbers. Each point in this plane represents a complex number expressed as \(z = x + yi\). Here, \(x\) is the real part, and \(y\) is the imaginary part.
  • The horizontal axis is the real axis.
  • The vertical axis is the imaginary axis.
Understanding the complex plane is critical when dealing with complex mappings. This allows you to visualize how transformations affect complex numbers. When a complex number undergoes a transformation, like squaring, its position on the plane changes.
Transformation
Transformation in mathematics involves changing a shape or a position. In the complex plane, transformations apply to complex numbers. They change how these numbers are plotted.
The particular transformation in our case is achieved through the function \(w = z^2\). This means we square the complex number \(z = x + yi\), resulting in \(w = x^2 - y^2 + 2xyi\). What does this do?
  • Real Part: From \(x^2 - y^2\), showing horizontal movement.
  • Imaginary Part: From \(2xy\), showing vertical movement.
Transformations like this are fascinating because they can warp the entire layout of points. They transition straight lines, circles, and other shapes into entirely new forms, showing the beauty of complex mappings.
Imaginary Axis
The imaginary axis is like the y-axis on a standard coordinate plane. It represents all numbers with no real parts, only imaginary.
In our exercise, after applying the transformation, the output \(w = 2x^2i\) lands entirely on the imaginary axis. The absence of a real part (where \(u = 0\) in \(w = u + vi\)) confirms this.
  • The numbers \(w\) plotted here follow the form \(vi\).
  • In our case, each \(w\) looks like \(2x^2i\). This means the transformation gives a purely imaginary result.
Transformations can relocate points within the complex plane from one axis to another or anywhere in between, shaping fascinating graphic representations.
Parabola
A parabola is a curve where each point is at an equal distance from a fixed point (the focus) and a line (the directrix). In our context, it emerges when we map points through the transformation.
Upon applying \(w = z^2\) to the line \(y = x\), we get results that resemble a parabola in form. Here the vertices touch the imaginary axis:
  • The transformation outputs \(w = 2x^2i\).
  • The imaginary part of \(w\) is proportional to \(x^2\), which is quadratic.
This behavior of complex mapping through squaring forms a parabola, famously opening along, and tracing the imaginary axis. It is disguised in the equation \(v = 2u^2\) in complex form, fixed where \(u=0\). This parabola displays how complex mappings create distinct geometrical shapes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the image of the given set under the reciprocal mapping \(w=1 / z\) on the extended complex plane.the semicircle \(|z|=\frac{1}{2}, \pi / 2 \leq \arg (z) \leq 3 \pi / 2\)

Find the image of the given set under the reciprocal mapping \(w=1 / z\) on the extended complex plane.the circle \(\left|z+\frac{1}{4}\right|=\frac{1}{4}\)

(a) Is it true that \(\lim _{z \rightarrow z_{0}} \overline{f(z)}=\lim _{z \rightarrow z_{0}} f(\bar{z})\) for any complex function \(f ?\) If so, then give a brief justification; if not, then find a counterexample. (b) If \(f(z)\) is a continuous function at \(z_{0}\), then is it true that \(\overline{f(z)}\) is continuous at \(z_{0}\) ?

Find the image of the given set under the reciprocal mapping \(w=1 / z\) on the extended complex plane.the line \(x=\frac{1}{6}\)

Find four sets in the complex plane that map onto the circle \(|w|=4\) under the mapping \(w=z^{4}\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.