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Let $\mu$ be a function on the complex plane with the property $\mu(z) = \overline{\mu(\bar{z})}$, such that $\mu(z) = \epsilon e^{-2\pi i \bar{z}}$ on the upper-half plane, where $\epsilon$ is a complex number such that $|\epsilon|<1$.

I'm interested in the solution to the following Beltrami equation with this Beltrami coefficient:

\begin{equation} \mu \frac{\partial f}{\partial z} = \frac{\partial f}{\partial \bar{z}} \end{equation}

Now, since $|\mu| = |\epsilon| \epsilon|e^{-2\pi y} < 1$ , (since $y >0$), there exists a unique quasiconformal solution $f$, satisfying $f(\bar{z})=\overline{f(z)}$ and fixing the points 0, 1, and infinity.

I'm basically trying to understand how $f_z$ grows.

Now, because $\mu$ is such a simple function, I feel like one should be able to explicitly write a solution in this case. I've tried many different approaches, but I have been very unsuccessful. I tried looking in Ahlfors and a few other references, but I haven't been able to find anything too useful.

Any ideas?
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Let $\mu$ be a function on the complex plane with the property $\mu(z) = \overline{\mu(\bar{z})}$, such that $\mu(z) = \epsilon e^{-2\pi i \bar{z}}$ on the upper-half plane, where $\epsilon$ is a complex number such that $|\epsilon|<1$.

I'm interested in the solution to the following Beltrami equation with this Beltrami coefficient:

\begin{equation} \mu \frac{\partial f}{\partial z} = \frac{\partial f}{\partial \bar{z}} \end{equation}

Now, since $|\mu| = |\epsilon| < 1$, there exists a unique quasiconformal solution $f$, satisfying $f(\bar{z})=\overline{f(z)}$ and fixing the points 0, 1, and infinity.

I'm basically trying to understand how $f_z$ grows.

Now, because $\mu$ is such a simple function, I feel like one should be able to explicitly write a solution in this case. I've tried many different approaches, but I have been very unsuccessful. I tried looking in Ahlfors and a few other references, but I haven't been able to find anything too useful.

Any ideas?
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Solving the Beltrami Equation for a very simple Beltrami Coefficient

Let $\mu$ be a function on the complex plane with the property $\mu(z) = \overline{\mu(\bar{z})}$, such that $\mu(z) = \epsilon e^{-2\pi i \bar{z}}$ on the upper-half plane, where $\epsilon$ is a complex number such that $|\epsilon|<1$.

I'm interested in the solution to the following Beltrami equation with this Beltrami coefficient:

\begin{equation} \mu \frac{\partial f}{\partial z} = \frac{\partial f}{\partial \bar{z}} \end{equation}

Now, since $|\mu| = |\epsilon| < 1$, there exists a unique solution $f$, satisfying $f(\bar{z})=\overline{f(z)}$ and fixing the points 0, 1, and infinity.

I'm basically trying to understand how $f_z$ grows.

Now, because $\mu$ is such a simple function, I feel like one should be able to explicitly write a solution in this case. I've tried many different approaches, but I have been very unsuccessful. I tried looking in Ahlfors and a few other references, but I haven't been able to find anything too useful.

Any ideas?