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Maybe an easy topology excercise. Say u is a weakly conformal map from a region of complex plane C to C. Then $u_z*{\bar u}_z=0$. How to derive that u is holomorphic or antiholomorphic, i.e. $u_z=0$ or $\bar u_z=0$ globally?

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seems enough to show that u is analytic? –  zalver Sep 3 '11 at 3:14
Found a proof. Show u is harmonic. –  zalver Sep 3 '11 at 5:19

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