I'm having trouble figuring out this question. $|z|=1$ and is on the contour which is a unit circle. Am I thinking about this too hard or is there a trick to doing this?
Show $x = \frac{1}{2}(z+\frac{1}{z})$ if $z = x+iy$
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complex-analysis
complex-numbers
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4If $|z|=1$ then $\dfrac{1}{z}=z^*$, right? – 2017-02-10
1 Answers
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We have for $0 \neq|z|=z\hat{z}=1$ $$ \frac{1}{z}z=z\frac{1}{z}=1 $$ as well as: $$ \hat{z}z=z\hat{z}=1 $$ Since the inverse is unique in $\mathbb{C}^*$, we can conclude that $\hat{z}=\frac{1}{z}$. Now we subsitute: $$ \frac{1}{2}(z+\frac{1}{z})=\frac{1}{2}(z+\hat{z})=\frac{1}{2}(x+iy+x-iy)=\frac{1}{2}2x=x $$ Edit: Also an easy way to see the important identity is: $\frac{1}{z}=\frac{\hat{z}}{\hat{z}z}=\frac{\hat{z}}{1}=\hat{z}$
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0You have a $y$ that should be an $x$. – 2017-02-10