Derivative of a complex integral function

Question

Let $\Omega$ be an open subset of $\mathbb{C}$ and let $z_0 \in \Omega$, $\rho\geq 0$ such that $$ U = \{ z_0 + x + iy \mid |x|\leq \rho, |y|\leq \rho \} \subset \Omega. $$ Suppose $f$ is an holomorphic function on $\Omega$. For $w = z_0 + x +iy \in U$ we define $$ F(w) = \int_0^x f(z_0+t)\, dt + i\int_0^y f(z_0 + x + it)\, dt.$$ I want to show that $\partial_xF(w) = f(w)$ and $\partial_yF(w)=if(w)$. I tried to rewrite both integrals in $F$ as integrals with parameters but I doesn't seem to work.

Answer 1

Since $f$ is holomorphic on $U,$ $$F(w)=\int_{z_0}^wf(z)dz$$ (along any path in $U$) hence $$F'=f.$$ On the other hand, by the Cauchy–Riemann equations, $$F'=\partial_xF=-i\partial_yF,$$ whence your two equalities.