Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-27, make an analysis as suggested in the summary above and then sketch the graph. $$ R(z)=z|z| $$

Step-by-Step Solution

Answer

The graph of \( R(z) = z|z| \) is a mapping of each complex number to a point on a circle of squared radius with the same angle.

Step 1: Understand the Function

The function given is \( R(z) = z|z| \). Here, \( z \) is a complex number, and \( |z| \) denotes the magnitude or modulus of the complex number \( z \). So, when we refer to \( z = x + yi \), where \( x \) and \( y \) are real numbers, the modulus is given by \( |z| = \sqrt{x^2 + y^2} \).

Step 2: Simplify the Expression

Rewrite the given function using the polar form of complex numbers. For any complex number \( z = re^{i\theta} \), where \( r = |z| \), the product can be expressed as \( R(z) = z|z| = re^{i\theta}r = r^2e^{i\theta} \). This indicates the transformation \( z \rightarrow r^2e^{i\theta} \), meaning the magnitude of \( z \) is squared, and the direction (angle) \( \theta \) remains unchanged.

Step 3: Sketch the Graph

In the complex plane, the transformation \( R(z) \) maps each point \( z \) to a new point with a magnitude squared but retaining its original direction. If the initial point \( z \) lies on a circle of radius \( r \), \( R(z) \) will map it onto a circle of radius \( r^2 \), maintaining the angle \( \theta \). To sketch the graph, visualize each point \( z \) and its corresponding image \( R(z) \) both on concentric circles having radii \( r \) and \( r^2 \), respectively.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to visualize complex numbers. Each point on this plane represents a complex number. A complex number is typically expressed in the form \( z = x + yi \), where \( x \) and \( y \) are real numbers. In this context:

• \( x \) is the real part of the complex number.
• \( y \) is the imaginary part of the complex number, with \( i \) being the imaginary unit such that \( i^2 = -1 \).

Each point \((x, y)\) in the complex plane corresponds to the complex number \( x + yi \). The plane itself is often called the Argand plane, named after J. R. Argand. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. Together, they allow us to effectively plot and manipulate complex numbers visually.
When dealing with functions of complex variables, like \( R(z) = z|z| \), visualizing them on the complex plane helps us understand the transformations involved.

Modulus of Complex Number

The modulus of a complex number gives us the size or length of the vector represented by the complex number in the complex plane. Given a complex number \( z = x + yi \), its modulus is calculated as:\[|z| = \sqrt{x^2 + y^2}\]This formula mirrors the Pythagorean theorem, where \( x \) and \( y \) are the lengths of the horizontal and vertical sides of a right triangle, respectively. The modulus corresponds to the hypotenuse, or the distance from the origin \((0,0)\) to the point \((x,y)\).
The modulus is always a non-negative number, as it represents a distance. This concept is crucial in understanding how complex numbers are scaled in different transformations. For instance, in the function \( R(z) = z|z| \), the modulus \(|z|\) scales the complex number \( z \), affecting its distance from the origin but not its direction.

Graph Transformation

Graph transformation in the context of complex functions involves mapping points from one location in the complex plane to another based on a given mathematical rule. With the function \( R(z) = z|z| \), this transformation modifies the distance of each point from the origin, though the angle (direction) remains constant.

• The transformation \( R(z) \) equates to multiplying the magnitude of \( z \) by itself, effectively squaring the modulus: \( |z| \rightarrow |z|^2 \).
• This means that a complex number with initial modulus \( r \) will be mapped to a new point at a distance \( r^2 \) from the origin, preserving its original angle \( \theta \).
• For instance, if a point lies originally on a circle of radius \( r \), it will move to a circle of radius \( r^2 \) after applying the transformation.

Graph transformations such as this demonstrate how complex functions can create dramatic changes in the graph, with areas expanding or contracting based on the operations applied to the modulus. These transformations help illustrate the powerful capabilities and applications of complex numbers in mathematics.