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September 7th, 2014, 06:15 AM  #1 
Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2  partial f(z bar)/partial z = partial f(z bar) / partial z bar, this cannot be right?
Hello there, I let $\sigma(z) = \overline{z}$ and $g: \Omega \to \mathbb{C}$. We let $f$ be some real differentiable function such that $g(z) = f(\overline{z})$. Then I found the following, which cannot be true. \[ \frac{\partial g(z)}{\partial z} = \frac{\partial f(\overline{z})}{\partial z} = \frac{1}{2}\left( \frac{\partial f(\overline{z})}{\partial x} i \frac{\partial f(\overline{z})}{\partial y} \right) = \frac{1}{2}\left( \frac{\partial f}{\partial x}\frac{\partial \sigma}{\partial x}  i \frac{\partial f}{\partial y}\frac{\partial \sigma}{\partial y} \right) = \frac{1}{2}\left( \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} \right) = \frac{\partial f(\overline{z})}{\partial \overline{z}} \] The most counterinutative problem is that $\frac{\partial f(\overline{z})}{\partial z} = \frac{\partial f(\overline{z})}{\partial \overline{z}}$ for all $f$, which sounds like nonsense. What is wrong? Thank you for your time. Kind regards, Marius 
September 7th, 2014, 06:25 AM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus 
Is σ analytic?

September 7th, 2014, 07:00 AM  #3 
Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2  
September 7th, 2014, 07:22 AM  #4 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus 
You are taking the composition of an analytic with a non analytic function.

September 7th, 2014, 07:39 AM  #5 
Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2 
How do you know that $f$ is analytic? Its real differentiable, but not holomorphic, hence not analytic, so neiher $f$ nor $\sigma$ are analytic. Though, why would it be a problem if one of them were not analytic whilst the other was? Thanks. Marius 

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bar, bar or partial, partial 
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