An exercise from Steins's complex analysis. This is an exercise from Steins's complex analysis chapter $8$:
Suppose $F(z)$ is  holomorphic near $z=z_0$ and $F(z_0)=F'(z_0)=0$, while $F''(z_0)\neq 0$.show that there are two curves $\Gamma_1$ and $\Gamma_2$ that pass through $z_0$ , are orthogonal at $z_0$ ,and so that $F$ restricted to $\Gamma_1$ is real and has a minimum at $z_0$ ,while $F$ restricted to $\Gamma_2$ is also real but has a maximum at $z_0$.
This hint is also given:
Write $F(z)=(g(z))^2$ for $z$  near $z_0$ , and consider the mapping $z \rightarrow g(z)$ and its inverse.
I really have no idea.
Thanks
 A: Suppose $f(z)=\left(g(z)\right)^2$. ($g(z)$ is well-defined in a disk centered at $z_0$ as $g$ is not constant).
Twice differentiating $f$ gives us:
$\begin{align}
&g(z_0)=0\\
&f''(z_0)=2\left[(g'(z_0))^2+g(z_0)g''(z_0)\right]\Rightarrow g'(z_0)\neq 0
\end{align}$
Hence by Exercise 1 (local bijection equivalence) of that chapter, $g$ is conformal in some disk around $z_0$. Therefore, $g^{-1}$ is also conformal.
Define these curves:
$\left\{\begin{array}{lc}
\gamma_1(t)=t\:;\:t\geq0 & \& &\Gamma_1=g^{-1}(\gamma_1(t))\:;\:t\geq0\\
\gamma_2(t)=it\:;\:t\geq0 & \& &\Gamma_2=g^{-1}(\gamma_2(t))\:;\:t\geq0\\
\end{array}\right.$
$\Longrightarrow\left\{\begin{array}{lc}
F(\Gamma_1(t))=t^2\:;\:t\geq0\\
F(\Gamma_2(t))=-t^2\:;\:t\geq0\\
\end{array}\right.$
Now $\Gamma_1$ and $\Gamma_2$ satisfy the problem conditions.
A: Fardad's answer is correct except for its explanation why $g(z)$ is well-defined. After all, we know $f(z) = z$ does not have a well-defined square root function around 0 as the square root would be multi-valued. Below is how to respond to this objection.
We can write $F(z) = (z-z_0)^2 h(z)$, where $h(z)$ is a nowhere vanishing holomorphic function around $z_0$, because $F(z_0) = F'(z_0) = 0$ and $F''(z_0) \neq 0$.
Since $h(z)$ is nowhere vanishing in a sufficiently small disc around $z_0$, $h^{1/2}(z) \equiv e^{\frac{1}{2}\log(h(z))}$ is well defined. (See Theorem 6.2 in Chapter 3.) Now we can define $g(z) \equiv (z-z_0)h^{1/2}(z)$.
