Complex chain rule applying to analytic function

Question

Can someone explain please what the complex chain rule (especially for harmonic functions) looks like? I thought it was this: If $F(\zeta)=f(\phi(\zeta))$, then $\frac{\partial F}{\partial \zeta}=\frac{\partial f}{\partial z}\frac{\partial \phi}{\partial \zeta}+\frac{\partial f}{\partial \bar{z}}\frac{\partial\bar{\phi}}{\partial \zeta}$. Is that correct? Further, my textbook says that is $\phi$ is analytic, then $\phi_{\bar{\zeta}}=0$. Do they mean that $\frac{\partial\bar{\phi}}{\partial \zeta}=0$? Why is that the case?

Answer 1

note that if $f(z) = f(x+iy)$ be analytic then:
$$2\frac{\partial f}{\partial z} = \frac{\partial f}{\partial x} + \frac{1}{i}\frac{\partial f}{\partial y}$$
$$2\frac{\partial f}{\partial \overline z} = \frac{\partial f}{\partial x} - \frac{1}{i}\frac{\partial f}{\partial y}$$ so answer to your second question follows by Cauchy-Riemann equations and the fact mentoned above.
Also chain rule in the sense of complex functions is:
consider $F(\phi(z))$ and assume $F$ and $\phi$ are both analytic at $z_0$. then
$$\frac{\partial F}{\partial z}(z_0) = \phi^{'}(z_0)f^{'}(\phi(z_0))$$