I have two very very basic question about some proof.
Theorem: A $C^1$ function $F$ is a primitive for $f(z) ~dz$ if and only if $F'=f$.
Proof: $F$ is a primitive of $f(z) ~dz$ $\Leftrightarrow dF = F_z ~dz + F_{\overline z } ~ d\overline z = f(z) \Leftrightarrow$ $$ \begin{align*} F_{\overline z} &= 0 \\ F_z &= f(z) \end{align*} $$ But (here comes the two questions) $F_z =F'$ so $F'= f(z)$.
The two questions are:
(1) How can the author guarantee that $F$ is holomorphic (has derivative $F'$)?
(2) If $F'$ exists, it is always true that $F_z = F'$. Remember that $F_z$ is defined by: $F_z = \frac12 (F_x + iF_y)$.