On page 151 of Hubbard's book, the author is proving the following theorem( Prop.4.6.2 ):
Suppose $\mu$ is a real analytic function on a domain $U$ of $\mathbb{C}$. Then every $z \in U$ has a neighbourhood $V$ in $U$ such that there is a real analytic function $f: V \to \mathbb{C}$ that is a homomorphism onto its image and satisfies the Beltrami equation: $$ \frac{\partial f}{\partial \bar{z}} = \mu \frac{\partial f}{\partial z} $$
Now, he approaches the problem in the following way. First, he rewrites the Beltrami equation in the following form: $$ (1-\mu(x,y))\frac{\partial f}{\partial x} + i(1+\mu(x,y))\frac{\partial f}{\partial y} = 0 $$
Doing this, he considers that $(x,y) \in \mathbb{C}^{2}$.The above equation is considered in a neighbourhood $W$ of $z_0 = (x_0,y_0)$ in $\mathbb{C}^2$ where $\mu$ is assumed analytical. Now there are some things that he says that I don't understand.
He considers the following differential equation: $$ \frac{dy}{dx} = i\frac{1+\mu(x,y(x))}{1-\mu(x,y(x))} $$ But I don't know if there is a theorem in Complex analysis which is analogous of Picard's about ODEs
He says that the functions which satisfy Beltrami equation are functions that are constant on the solution of the ODE. I thought about differentiating the equation $(x,f(x))$ along the curve of the solution of ODE. However, he than says that this is a brilliant idea of Gauss and I don't see what is idea or why should it be brilliant. I don't see how solving this ODE gives us solutions of the initial problem.
He than uses inverse function theorem for the function which is $y$ on the line $x = x_0$ and says: $$ \frac{\partial f}{\partial x}(x_0,y_0) = -i \frac{1 + \mu(x_0,y_0)}{1 - \mu(x_0,y_0)} $$
He says that the above number is not real( another way of saying that it is not zero? ) and applies inverse function theorem and says that f induces local diffeomorphism between $W \cap \mathbb{R}^2$ and $\mathbb{C}$. What I don't see is why is it $W \cap \mathbb{R}^2$ and not just $W$?
