A few complex number problems

**Lemons123** · January 3rd 2012, 08:19 AM

I have attempted to solve it but arrive at either no solution or a solution much different from the textbook's.

1. If w = $\frac{z + 1}{z - 1}$ and |z| = 1, find Re(w).

I plugged in a + bi for z and attempted to simplify.

**FernandoRevilla** · January 3rd 2012, 09:08 AM

Use that $\textrm{Re}(w)=\frac{1}{2}(w+\bar{w})$ and $|z|^2=z\bar{z}=1$ .

**Opalg** · January 3rd 2012, 12:50 PM

Originally Posted by Lemons123

I have attempted to solve it but arrive at either no solution or a solution much different from the textbook's.

1. If w = $\frac{z + 1}{z - 1}$ and |z| = 1, find Re(w).

I plugged in a + bi for z and attempted to simplify.

If If $w=\frac{z + 1}{z - 1}$ then $w(z-1) = z+1.$ Solve that for z to get $z = \frac{w+1}{w-1}.$ If $|z|=1$ it follows that $|w+1| = |w-1|.$ Thus w is equidistant from 1 and –1, and so Re(w) = 0.

**FernandoRevilla** · January 3rd 2012, 02:04 PM

Or (using $z\bar{z}=1$ ) :

$\textrm{Re}(w)=\frac{1}{2}(w+\bar{w})=\frac{1}{2} \left(\frac{z+1}{z-1}+\frac{\bar{z}+1}{\bar{z}-1} \right)=\frac{1}{2}\dfrac {0}{(z-1)(\bar{z}-1)}=0$

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