The problem statement is:
Use conformal mapping to find a harmonic function $U(z)$ defined on the unit disc $∣z∣<1$ such that
$$\lim_{r→1} U(re^{iθ}) = \left\{\begin{aligned} &+1 &&: 0<\theta<\pi\\ &-1 &&: \pi<\theta<2\pi \end{aligned} \right.$$
Give the correct determination of any multiple-valued functions appearing in your answer.
How is my work?
First, I took the mapping $w= \large \frac{z-i}{z+i}$, and solved for z.
Now I have a mapping from the unit disk to the UHP. The mapping is
$$ h := \frac {-i(1+w)}{w-1}$$
Then arg(h) is 0 on $R^+$, which happens to be the image of the lower semi-circle, and arg(h) = $\pi$ on $R^-$, which is the image of the upper semi-circle.
And finally take $ cos(arg(h))$ gives -1 on the upper semi-circle and +1 on the lower semi-circle, so we negate this function to get
$$\Gamma:= -cos(arg(h)),$$ $$= -cos(arg(\frac {-i(1+w)}{w-1}))$$
which gives us what we want.
We note that both the arg and cosine functions are harmonic, since cosine is the real part of $e^{i\theta}$ and arg(z) is the imaginary part of log(z) (say, cutting away the negative imaginary axis.)
Any comments or suggestions are welcome.
Thanks,
Hint: it's not hard to find a harmonic function on the upper half plane with boundary values $+1$ on one side of the $x$ axis and $-1$ on the other. Find a conformal map of that to the disk...