The picture shows what the function f: $\mathbb{C}\to\mathbb{C}\cup\infty$ does to the plane.
The values 0 at 0, 1 at $\pm$1, and $\infty$ at $\pm i$ are specified.
To elaborate on the picture: there is a unit disk, with a plus sign on the part of the disk in the first quadrant, a minus sign on the part of the disk in the second quadrant, plus sign for the third quadrant and minus sign on the fourth quadrant.
Outside this unit disk, there is a minus sign on the first quadrant, plus sign on the second quadrant, minus sign on third quadrant, and plus sign on the fourth quadrant.
Now, back to the question:
The signatures +/- indicate that the regions so marked are mapped 1 to 1 onto the upper / lower half-plane.
What is f? Explain why it cannot be otherwise.
Edit:
I'd like to post an old solution to this problem, as well as ask for help in the intermediate steps that were implicit in the solution itself:
One solution offered by a student was this:
Using the Reflection Principle, the function f is entirely determined by where it maps a single quarter-circle sector, say, the quarter-circle sector in the first quadrant.
Question no. 1: Why do we use the Reflection Principle? Is it because, say, the first quarter-circle sector has a + sign, so f maps it to the upper half plane, and the Reflection Principle somehow tells us that f maps the rest of the first quadrant (outside of the quarter-circle sector), which has a minus sign, to the lower half plane? If this is the case, how do we actually know the conditions are fulfilled to actually apply the Reflection Principle? Do we think of the boundary of this quarter-circle sector to be the real line, where f takes real values and is continuous on this boundary? How can we accept that? Do we make some sort of intermediate transformation so that the image of the boundary is the real line? (or perhaps just…real-valued, not necessarily the entire real line?)
If we consider the mappings g(z)=z^2 and h(w)=−w−1/w, this quarter-circle is mapped to a semicircle and then to the upper half plane, both mappings invertible.
Question no. 2: How do we know to immediately consider these two specific mappings? Why not consider other mappings of the quarter-circle sector in the first quadrant? What makes these two choices of mappings so convenient?
Thus our mapping must be of the form f(z)=T∘h∘g where T maps from the upper half plane onto the upper half plane. Such a map must be a linear fractional transformation, a consequence of Maximum Modulus Principle / Schwarz' Lemma.
Question no. 3: At which point did we use the Maximum Modulus Principle and Schwarz's Lemma?
To determine which T it is, we consider the mappings now. $h(g(0))=\infty$,$h(g(i))=2$,$h(g(1))=−2$, so T must map $\infty$ to 0, 2 to $\infty$, and −2 to 1, so $$T(z)=−4/(z-2)$$ and putting the maps together
$$f(z)=4z^2/(1+z^2)^2$$
Question no. 4: How do we know T will map the upper half plane to the upper half plane? It seems that we only know what T does…to 3 specified pre-image points.
Apparently, this final mapping is the mapping that describes the action indicated in the picture, which gives only + and – signs inside the unit disk and outside of it.
Any help would be greatly appreciated.
Thanks in advance,
Best Answer
Q1. Reflection is a natural thing to use on any analytic boundary and there is no difference between the circle and the real line because you can always apply a fractional-linear transformation to map one to another. The reason the reflection principle is stated for the line in the books is merely that the formula for the line is the neatest one of all formulae of that type.
Q2. Do you know any other way to conformally map the quarter circle to the upper half-plane? The guy liked Joukovsky, of course, but reducing the number of boundary lines from 3 to 2 is an inevitable first move (which is what $z^2$ does), after which a less sophisticated person would just move one of the ends of the diameter to $\infty$ by a fractional linear map to get a quadrant, which can be opened to a half-plane by another squaring. Note however that this approach would move one of the points that has to stay finite to infinity and you would need to bring it back from there afterwards by an extra fractional linear mapping.
Q3 and Q4. What is really going on here is that you should know the fractional linear mappings of the upper half-plane (or, if you prefer, the unit disk) to itself and the proof of their full classification (which uses the Schwarz lemma) together with the result that any three points on the boundary can be moved to any other three points with the same orientation, but then the mapping is determined uniquely. Apparently, the guy just referred to all that en passe to explain what was going on but without trying to make each reference as precise as possible (which is normal when both people conversing share common allusions and associations but which may make the conversation incomprehensible for a third party). So, just reread those sections in your textbook or notes.