Let $z$ be a complex number, set $w =z+1/z-1$

For what values of $z$ is the real part of $w$ positive?

I have tried working with the formula $2\Re(w) = z + \bar z$ but have not been able to get anywhere. Can someone help?


closed as off-topic by Did, Claude Leibovici, Qwerty, DonAntonio, tatan Sep 22 '16 at 11:44

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  • $\begingroup$ Let $z=x+y\mathrm{i}$. $\endgroup$ – Colescu Sep 22 '16 at 5:20
  • $\begingroup$ I have tried working with the formula ... The formula you posted is wrong. That should be $2 \Re(w) = w + \bar w$ instead. Try working with that one. $\endgroup$ – dxiv Sep 22 '16 at 5:42
  • 1
    $\begingroup$ Is it $\;\frac{z+1}{z-1}\;,\;\;or\;\;z+\frac1z-1\;,\;\;or\;\;z+\frac1{z-1}\ldots...??$ Use some parentheses! $\endgroup$ – DonAntonio Sep 22 '16 at 7:52
  • $\begingroup$ @DonAntonio it is (z+1)/(z-1) $ $\endgroup$ – jmsac Sep 22 '16 at 15:30

HINT: Let $z=a+bi$ where $a$ and $b$ are real numbers. Then $w=a+bi+1/(a+bi)-1=a+bi+(a-bi)/|z|^2-1.$


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