I've tried this problem for awhile now but something strange is happening at the end of my solution.
The question asks to map the region between $|z| =1$ and $|z-\frac{1}{2}| = \frac{1}{2}$ on a half plane.
My work, in a nutshell is:
a) look at the mapping $w(z) = \frac{z}{z-1}$. This map has a pole at 1, which lies on both circles. So, under this mapping, both circles will map to straight lines.
b) also, this map fixes the point 0.
c) I test some points on both pre-image circles and determine exactly where the two lines are.
d) I get an infinite vertical strip, with the left boundary line = the imaginary axis, the right boundary line = $Re(z)=1/2$.
Here's where I seem to be going wrong:
e) I rotate this vertical strip by an angle of $\pi/2$, so I multiply the map by the factor $e^i.\frac{\pi}{2}$, getting
$w(z)= e^{(i\pi/2)(z/(z-1))}$, which is just $i.\frac{z}{z-1}$.
If this is correct, now I should have a horizontal strip, with lower boundary line = real axis, and upper boundary line = $Im(z) = i/2$.
f) Now, I scale by a factor of $2\pi$, so my map is $2\pi i.\frac{z}{z-1}$. This keeps the lower boundary line the same (on the real axis), but the upper boundary line is now $Im(z) = i\pi$.
Finally, to fill up the entire upper half plane, apply the exponential map to $w(z)$, getting:
$$
w(z) = exp\left( 2\pi i.\frac{z}{z-1}\right).
$$
But this is weird, because now this mapping is $w(z) = (e^{2\pi i})^{z/(z-1)} = 1^{z/(z-1)} = ….1$?
Though, if I just stare at this horizontal strip, and notice that the lower boundary (real axis) contains points of the form $x+i0$, and applying the exponential map to these points gives images of the form $e^x$, which is always positive. So, the lower boundary line (real axis) shifts over to the positive real axis. Similarly, the upper boundary line, $Im(z) = i\pi$, contains points of the form $x+i\pi$, and applying the exponential map to these points gives images of the form $e^{x+i\pi} = e^x . e^{i\pi} = -e^x$, which is always negative. So, it seems that the upper boundary line shifts down and over to the negative real axis. I'm guessing at this point that the job is done, and I have mapped the original region between the two circles to the upper half plane.
Intuitively, this sounds reasonable, but I'm not sure why I'm getting a map of $w(z) = 1$.
Any help would be greatly appreciated.
Thanks,
Best Answer
Your solution is correct.
But the reasoning
But this is weird, because now this mapping isis wrong. Any $z \neq 0$ can be written as $z=e^{\log z}=e^{2\pi i \frac{\log z}{2\pi i}}$. Following your reasoning, we would have $z=1$ for any $z\neq 0$.The reason is that the definition $a^b$ is quite different in complex analysis. You have to define it as $e^{b \log a}$. But logarithm is multi-valued. Therefore you need to be very specific which branch you are using.
Intuitively speaking, when you write down $e^{2\pi i z} = 1^z$, you also have $1^z= e^{z\log 1}$. But here we have to take $\log 1 = 2\pi i$ instead of $\log 1 = 0$ for consistency.