Find the condition where the transformation z maps the unit circle in the w-plane into a straight line

I. Introduction

In the field of complex analysis, a transformation is a function that maps one set of points to another. Specifically, a transformation in the complex plane maps points in the z-plane to points in the w-plane. The unit circle in the z-plane is defined as the set of all points z such that |z| = 1. A straight line in the w-plane can be defined by a linear equation in the real and imaginary parts of w.

II. Mathematical Background

A conformal mapping is a function that preserves angles locally. In other words, if two curves intersect at a certain angle in the z-plane, their images under the conformal mapping will intersect at the same angle in the w-plane. A Mobius transformation, also known as a linear fractional transformation, is a specific type of conformal mapping that is defined by the equation w = (az + b) / (cz + d), where a, b, c, and d are complex numbers and ad - bc ≠ 0.

III. Derivation of the Condition

For a transformation to map the unit circle in the z-plane to a straight line in the w-plane, the denominator of the Mobius transformation must be zero along the unit circle. This implies that cz + d = 0 for all z such that |z| = 1. Solving this equation for z gives z = -d/c, which lies on the unit circle if and only if |d/c| = 1. This condition can be satisfied if c ≠ 0 and d = -a. Therefore, the condition for the transformation to map the unit circle to a straight line is that c ≠ 0 and d = -a.

This condition implies that the Mobius transformation takes the form w = z / (cz - a), which is a linear fractional transformation with a zero denominator. This type of transformation maps the unit circle in the z-plane to a straight line in the w-plane.

IV. Example

Consider the transformation w = z / (1 - z). This transformation satisfies the condition derived above, with c = 1 and a = -1. To see that this transformation maps the unit circle to a straight line, note that if z is on the unit circle, then |z| = 1 and hence |1 - z| = |z - 1| = 1. Therefore, |w| = |z / (1 - z)| = 1, which means that w lies on the unit circle in the w-plane. This shows that the transformation maps the unit circle in the z-plane to the unit circle in the w-plane.

A diagram illustrating this transformation would be helpful here. The diagram should show the unit circle in the z-plane and its image under the transformation in the w-plane, which is a straight line.

V. Conclusion

In conclusion, the condition for a transformation to map the unit circle in the z-plane to a straight line in the w-plane is that the transformation is a linear fractional transformation with a zero denominator. This condition can be satisfied if the coefficients of the transformation satisfy c ≠ 0 and d = -a. This type of transformation has potential applications in complex analysis and other fields, as it can simplify the analysis of functions and equations by transforming them into simpler forms.

Summary

A transformation in the complex plane maps points in the z-plane to points in the w-plane. To map the unit circle in the z-plane to a straight line in the w-plane, the transformation must be a linear fractional transformation with a zero denominator. The condition for this transformation is that the coefficients of the transformation satisfy c ≠ 0 and d = -a.

Analogy

Imagine a movie where the main character travels through a magical portal. In the z-plane, the main character is on the unit circle, representing their journey. When they enter the portal, they are transformed into the w-plane, where they become a straight line. The condition for this transformation is like a secret code that the main character must enter to successfully travel through the portal.