Is there a conformal mapping that sends the upper semi-circle to the positive (or negative) real line, and the real interval $[-1,1]$ to the other half of the real line?
I am considering the upper semi-circle $\{|z|=1, 0<arg(z)<\pi\}$, with the line $[-1,1]$ that closes the loop.
I want to map the arc to a half line, and the interval to the other half line.
Is this possible?
I have tried for a bit, with no success so far.
The end goal is to find a harmonic function on this upper semi-disk, but I think the conformal mapping needs to be executed first…
I used the Joukowski transform, since part of the problem statement specifically asks to compute its image, but this transform takes the arc to an interval $[-2,2]$, which is not at all helpful.
Thanks.
Best Answer
It’s not so hard. First use a fractional-linear that takes $1\mapsto0$, $-1\mapsto\infty$, $0\mapsto i$, and $i\mapsto1$. You get this by $$z\mapsto-i\frac{z-1}{z+1}\,.$$ You see that this therefore takes your semicircular arc to the positive real axis and the line segment $[-1,1]$ to the positive imaginary axis. Now square the result. Your mapping is $$z\mapsto-\left(\frac{z-1}{z+1}\right)^2\,.$$