Assuming either |z|=1 or |w|=1... basic complex analysis

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Hint: If $|z|=1$, then $\overline z=1/z$. Use this to show that $|z-w|=|1-\overline zw|$.

So, $$|1-\overline zw|=\left|1-\frac wz\right|=\left|\frac 1z\right||z-w|=|z-w|.$$

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Violet Rodriguez
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Violet Rodriguez

Updated on August 01, 2022

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  • Violet Rodriguez
    Violet Rodriguez over 1 year

    Assuming either $|z|=1$ or $|w|=1$ and $\bar z w$ $\neq 1$, prove that

    |$\frac{z-w}{1-\bar zw}$| $=1$

    Any hint? I don't have a clue

  • Tim Raczkowski
    Tim Raczkowski over 7 years
    This proof works of course, but it definitely is the long way. See my edited answer.