I am curious if conformal maps preserve structure. What I mean is suppose I have the unit disk $\mathbb{D}$ and that I am mapping it to the upper half plane. As I walk around the boundary of the circle will the image also be moving in a continuous fashion across the real axis, or is it allowed to jump around (i.e. is the person jumps sporadically from one place to another?
In other words, we know that $$f(z) = i\frac{1-z}{1+z}$$
is a conformal map from the unit disk to the upper half plane. Then we have that
$$f(i) = 1\\ f(-i) = -1 \\ f(1) = 0\\ f(-1) = \infty.$$
I interpret this as if I start from $i$ and walk counter-clockwise around the circle then my image will be starting from $1$ on the real axis and go toward infinity before coming back at $-1$ then at $0$ and then start from the beginning again. Is this generally the case?
The motivation behind this question is to solve PDEs using conformal maps, and hence I would like to know here my boundary values are being mapped to. Using the earlier example, if I have boundary values prescribed on part of the disk, I would like to know how this is mapped on the half plane.
Best Answer
If I understood your question correctly, the image of a mapping given by a continuous function $w=f(z)$, will map every point $z$ that lays in a curve or in a simply conected region to the $w$ complex plane continuously.
One could argue that mapping a bounded region in the $z$ plane with a bounded $f(z)$ (in moduli) in that region, the image will result in a bounded region in the $w$ plane.
In complex mappings we always work on the extended complex plane: $\mathbb{C}^*=\mathbb{C} \cup \{ \infty \}$. This makes all Moebius transformations defined everywhere, and also, continous everywhere in $\mathbb{C}^*$.
I know this isn't a rigurous proof, but I guess it works out for what you want to do. Your Moebius transformation will not make any jumps, but it could seem that it behaves "weird" when you reach the point in $z$ that gets mapped to $\infty$ (and let's be honest, the extended complex plane sometimes seems "weird").
Tell me if you find this useful.