I'm trying to prove that the function: $f(z)= x^2-y^2+2ixy$ is not conformal. Another student suggested that I should use the following: $x = \frac{z+ \overline{z}}{2}$ and $y = \frac{z - \overline{z}}{2}$, to help guide me towards a solution, but we haven't been explicitly taught this yet. So how do we know that $x = \frac{z+ \overline{z}}{2}$ and $y = \frac{z - \overline{z}}{2}$?
Complex Analysis Question: how to find x, y in relation to z, z bar
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complex-analysis
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0Have you tried substitution $z=x+iy$ and run through the arithmetic? – 2017-02-17
1 Answers
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We have $$z=x+iy$$ and $$\overline{z}=x-iy.$$ Adding these we obtain $$ z+\overline{z}=2x.$$ Similarly, substracting them we obtain $$ z-\overline{z}=2iy.$$
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0Thank you! I knew it was a simple connection that I was missing. – 2017-02-17