Word Count : 496

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To solve this problem, we will tackle it in two parts. First, we'll prove that the given linear fractional transformation \(\phi(z) = \frac{2z - 1}{2 - z}\) maps the unit circle \(|z| = 1\) into itself. Second, we'll prove that \(\phi(z)\) is conformal in the closed unit disk \(\overline{D} = \{z : |z| \leq 1\}\). ### Part 1: Prove that \(\phi(z)\) maps the unit circle \(|z| = 1\) into itself Let \(z\) be any point on the unit circle, meaning \(|z| = _______ ___ _______ ___ _________ ________ _______ ___ _______ _____.
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