Question

Consider the circle \(|z-1|=1\) a. Rewrite the equation in rectangular coordinates by setting \(z=\) \(x+i y\) b. Sketch the resulting circle using part a. c. Consider the image of the circle under the mapping \(f(z)=z^{2}\), given by \(\left|z^{2}-1\right|=1\) i. By inserting \(z=r e^{i \theta}=r(\cos \theta+i \sin \theta)\), find the equation of the image curve in polar coordinates. ii. Sketch the image curve. You may need to refer to your Calculus. II text for polar plots. [Maple might help.]

Short answer

The equation of the circle in rectangular coordinates is \((x-1)^2 + y^2 = 1\), while that of the image curve in polar coordinates is given by \(|r^2(2\theta)-1| = 1\). Sketching these on a cartesian or polar plane respectively can easily be achieved with the use of a software like Maple, as the sketches would require dynamic plotting tools.

Step-by-step solution

Transforming Complex Number Equation into Rectangular Coordinates

Given the complex number equation |z-1| = 1, replace z with its rectangular coordinate equivalent, x+iy. This yields the equation |x + iy - 1| = 1. Using the definition of absolute value, we get \((x-1)^2 + y^2 = 1\).

Sketching the Initial Circle

Using the equation from step 1, \((x-1)^2 + y^2 = 1\), we realize that this is the equation of a circle which can be sketched on an xy-plane. The circle, centered at (1,0) has a radius of 1.

Applying the Mapping Function

The given mapping function is f(z) = \(z^2\). So, replacing z in the given complex number equation with \(z^2\), we get \(|(z^2)-1| = 1\). Replacing z with \(re^{i\theta}\) in order to express it in polar coordinates, we get \(|(r^2 e^{2i\theta})-1| = 1\). Simplifying gievs us \(|r^2 (cos(2\theta)+isin(2\theta))-1| = 1\). Using the definition of absolute value in polar coordinates, we obtain \(|r^2(2\theta)-1| = 1\).

Sketching the Transformed Curve

The resulting equation from step 3 can be sketched in polar coordinates. A rough sketch of the curve can be drawn by first finding data points that satisfy the equation, then plotting these points, and connecting them in a smooth curve. We need to emphasize that the precise sketching process will likely require the use of software like Maple or a similar tool.

Key concepts

Rectangular Coordinates

Complex numbers can be represented in a form akin to traditional two-dimensional Cartesian coordinates, where the 'x' represents the real part and the 'y' stands for the imaginary component. This system is known as rectangular coordinates. In the context of our exercise, we translate the complex number equation by setting a value for the complex number 'z' as equal to the format of 'x + iy', where 'i' represents the square root of -1.

When we convert the given circular equation from its absolute form, \( |z-1| = 1 \), into rectangular coordinates, we replace 'z' with 'x + iy' to visualize the equation in a more familiar xy-plane. This treatment makes it more accessible for graphing and understanding the geometric representation of complex number equations. For the circle given in our example, the equation \( (x-1)^2 + y^2 = 1 \) tells us that we are dealing with a circle centered at (1,0) with a radius of 1, which can be conveniently sketched using these coordinates.

Polar Coordinates

Polar coordinates offer an alternative way to describe the position of points in a plane. Instead of using two perpendicular lines as references, like in rectangular coordinates, polar coordinates employ a single point known as the pole (or origin), and a ray emanating from that point called the polar axis. The position of a point is then determined by its distance from the pole, which is the radius 'r', and the angle 'θ' formed between the polar axis and the line connecting the point to the pole.

In our exercise, we transform the equation of a circle into polar coordinates by inserting the polar form of the complex number 'z', expressed as \( r e^{i\theta} \). This represents 'z' in terms of its magnitude 'r' and angle 'θ', as \( z = r (\cos \theta + i \sin \theta) \) for step c of the problem. Polar coordinates are valuable in visualizing and understanding the properties of complex numbers, particularly when dealing with rotational and radial symmetries.

Mapping Functions

In our context, mapping functions refer to the transformation of complex numbers from one form to another. The function \( f(z) = z^2 \) takes each complex number 'z' and maps it to another complex number by squaring it. This transformation affects both the magnitude and the angle of 'z'.

As suggested by the exercise, we apply the mapping function to the equation of the circle, which modifies its size and position. This is a critical concept because mapping functions can significantly change the geometric shapes represented by complex number equations. By understanding how these functions work, students can gain deeper insights into the dynamic nature of complex relationships in mathematics.

Absolute Value of Complex Numbers

The absolute value, or modulus, of a complex number, is a measure of its distance from the origin in the complex plane. It’s analogous to how we view magnitude in a two-dimensional space. In our exercise, the absolute value equation \( |z-1| = 1 \) determines a set of points forming a circle of radius 1 centered at the point corresponding to the complex number '1'.

This concept serves as the foundation for solving the given equation both in rectangular and polar coordinates. For our circle's equation, the absolute value provides the necessary condition for all points z that are uniformly one unit away from the complex number '1'. Understanding this concept is crucial for interpreting and sketching graphs of complex number equations, as it represents the core principle behind the shapes and curves these equations describe in the complex plane.