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Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S = { z: |z| = 1 }). Consider the function f(z) = z z* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane

1. unit circle
2. horizontal axis line segment from origin to (1, 0)
3. the point (1, 0)
4. the entire horizontal axis

Correct answer: the point (1, 0)

Solution

The function f(z) = z z* simplifies to |z|², which equals 1 for any point z on the unit circle. Since |z|² is always 1, f(z) evaluates to 1 for all z in S, mapping every point on the unit circle to the single point (1, 0) in the complex plane.

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