
Complex Analysis Complex Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
September 7th, 2014, 05:15 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, 05:25 AM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus 
Is σ analytic?

September 7th, 2014, 06:00 AM  #3 
Senior Member Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2  
September 7th, 2014, 06:22 AM  #4 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 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, 06:39 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 

Tags 
bar, bar or partial, partial 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
partial sum  sagicoh  Real Analysis  3  December 29th, 2012 09:28 AM 
Partial limits of cos(pi*n/3)  peripatein  Calculus  1  November 10th, 2012 08:23 AM 
Partial Sum  chapsticks  Calculus  5  March 10th, 2012 05:57 PM 
Partial sum of n^2  ZardoZ  Real Analysis  4  August 5th, 2011 07:41 AM 
partial derivates  riotsandravess  Calculus  1  February 13th, 2011 04:36 PM 