Mapping the Complex Region Defined by |z| ≤ 2 and 1/|z| ≤ 2
Mapping the Complex Region Defined by |z| ≤ 2 and 1/|z| ≤ 2
Understanding the mapping of complex regions in the complex plane is fundamental in complex analysis. This article will guide you through the process of mapping the region defined by the inequalities |z| ≤ 2 and 1/|z| ≤ 2. This involves understanding how these inequalities affect the complex plane and how to visualize them graphically.
The Initial Inequality: |z| ≤ 2
In the complex plane, |z| ≤ 2 represents a circle centered at the origin with a radius of 2. The inequality |z| represents the magnitude (or modulus) of the complex number z, which is expressed as:
|z| √(x^2 y^2)
For |z| ≤ 2, we have:
x^2 y^2 ≤ 4
The region defined by this inequality is a circle with the equation x^2 y^2 4 and includes everything inside this circle.
The Inequality 1/|z| ≤ 2
The inequality 1/|z| ≤ 2 means that the modulus of z must be greater than or equal to 1/2. This can be written as:
|z| ≥ 1/2
Graphically, this represents the exterior and boundary of a circle centered at the origin with radius 1/2. The region defined by this inequality is everything outside the circle with the equation x^2 y^2 1/4, including the circle itself.
Combining the Two Inequalities
Combining the inequalities |z| ≤ 2 and 1/|z| ≤ 2 requires us to find the intersection of the two regions:
|z| ≤ 2 and |z| ≥ 1/2
This results in a washer-shaped region (also known as an annulus) that lies between the circles with radii 1/2 and 2, including the boundaries of these circles.
Detailed Mapping Process
Let's go through the steps to map this region in detail:
Step 1: Expressing the Inequalities in Terms of z and 1/z
Given the initial form of the inequality:
1/|z| ≤ 2
We can rewrite this as:
|1/z| ≤ 2
This means:
|z| ≥ 1/2
Which can be written as:
|z| ≥ 1/|z|
This inequality represents the exterior and boundary of the circle with radius 1/2.
Step 2: Combining Both Inequalities
Combining the inequalities |z| ≤ 2 and |z| ≥ 1/2, we need to find the intersection of the two regions:
|z| ≤ 2 and |z| ≥ 1/2
This results in the region bounded by the two circles with radii 1/2 and 2, including the circles themselves.
Graphical Visualization
To visualize this region, consider the following:
Circle with Radius 2
The equation for the circle is:
x^2 y^2 4
This circle includes the region inside the circle, as well as the circle itself.
Circle with Radius 1/2
The equation for the circle is:
x^2 y^2 1/4
This circle only includes the region outside the circle, including the circle itself. To get the combined region, we need to take the area that is between these two circles.
Conclusion
The region defined by the inequalities |z| ≤ 2 and 1/|z| ≤ 2 is a washer-shaped region (annulus) that lies between x^2 y^2 1/4 and x^2 y^2 4, including the boundaries of these circles.