Partial derivative of the harmonic complex function

[Adel Makram]

For a harmonic function of a complex number $z$, $F(z)=\frac{1}{z}$, which can be put as $F(z)=f(z)+g(\bar{z})$and satisfies $\partial_xg=i\partial_yg$. But this function can also be put as $F(z)=\frac{\bar{z}}{x^2+y^2}$ which does not satisfy that derivative equation!

Sorry, I should have put this thread in homework section.

 

[Svein]

Actually, if F is analytic, $\frac{\partial F}{\partial\bar{z}}=0$.