Transformation of a Circle under w1/z: Interior to Exterior

In the realm of complex analysis, transformations of geometric figures play a significant role in understanding the structure and properties of these figures. One intriguing transformation is the mapping of a circle under the function (w frac{1}{z}). This transformation has unique properties that can be explored through both theoretical analysis and graphical visualization.

Introduction to Complex Transformation

A complex transformation is a function that maps points in the complex plane to other points in the same plane. These transformations can reveal deep connections and symmetries within the complex plane. One such transformation is given by the function (w frac{1}{z}), which is particularly interesting when applied to circles in the complex plane.

The Circle in Question: z1

The circle defined by (z1) represents a circle in the complex plane with a radius of 1 centered at the origin. This circle is a well-known and simple object, but when we apply the transformation (w frac{1}{z}) to it, we observe a fascinating phenomenon.

Transformation Properties

The transformation (w frac{1}{z}) has the property of mapping circles and lines to circles and lines in the complex plane. Specifically, it maps the interior of the circle (z1) to the exterior of a new circle, and vice versa. Let's explore this in more detail.

Mapping the Interior to the Exterior

Consider a point (z) inside the circle defined by (z1). This means that (|z| 1). When we apply the transformation (w frac{1}{z}), this point (z) is mapped to a point (w) in the complex plane such that (wfrac{1}{z}). Since (|z| 1), it follows that (left|frac{1}{z}right| 1). Therefore, the point (z) inside the circle is mapped to a point outside the circle (z1).

Mapping the Exterior to the Interior

Conversely, consider a point (z) outside the circle defined by (z1). This means that (|z| 1). When we apply the transformation (w frac{1}{z}), this point (z) is mapped to a point (w) in the complex plane such that (wfrac{1}{z}). Since (|z| 1), it follows that (left|frac{1}{z}right| 1). Therefore, the point (z) outside the circle is mapped to a point inside the circle (z1).

Points on the Circle

It's important to note that any point on the circle defined by (z1) is mapped to itself under the transformation (w frac{1}{z}). This is because if (|z|1), then (left|frac{1}{z}right|1). Therefore, the point (ze^{itheta}) (where (theta) is a real number) is mapped to (we^{-itheta}), which is the same point on the circle.

Graphical Visualization and Applications

To better understand the transformation, consider drawing the circle defined by (z1) and its image under the transformation (w frac{1}{z}). You will observe that the interior of the circle is mapped to the exterior of a new circle, and the exterior of the circle is mapped to the interior of the same circle. This transformation can be visualized using complex plane drawing software.

Further Exploration

The transformation (w frac{1}{z}) is an example of a M?bius transformation, which is a class of transformations that map circles and lines to circles and lines. Such transformations are fundamental in complex analysis and have applications in various fields, including fluid dynamics, conformal mapping, and geometric function theory.

Conclusion

The transformation of a circle defined by (z1) under the function (w frac{1}{z}) is a fascinating topic in complex analysis. It demonstrates the unique properties of M?bius transformations and reveals the deep interconnections within the complex plane. Understanding such transformations can provide valuable insights into the behavior of geometric figures in complex analysis.