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I need to find a harmonic function in the region ${ {z: |z|<1: Imz>0} }$ whose boundary values are in 1 on the interval $(-1,1)$ and $0$ on the half- circle.

I have no clue, where to start!

I do not think I can proceed like as I used to for finding conformal mapping.

I do not mind getting details since this one of the qual question. Any help much appreciated.

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up vote 2 down vote accepted

First, find a conformal map from the half-disk to the upper half plane, making sure that the half circle is sent to the positive real half-line, and the diameter of the half-disk is sent to the negative real half-line. (Hint: look at the map $z \mapsto - \frac{1}{2}(z + z^{-1})$). Let's call this map $\Phi$.

Second, consider the Arg function (or, if you like, the imaginary part of the logarithm with a suitable branch cut making it analytic on the upper half plane). Define your Arg so as to assign the value zero to the positive half line and $\pi$ to the negative half line.

Third, having done all this, you've concocted a harmonic function $u$ on the upper half plane. Now form $u \circ \Phi$, which takes on the correct boundary values.

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Would you please prove more rigorously please. I kind of lost in the second and third part. – Deepak Dec 27 '12 at 0:38

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