I'm looking at an old exam problem that shows a picture of what the function f does to the plane. On the upper right quadrant, there is a + sign, which indicates that f maps this quadrant one-to-one and onto the upper half plane. On the upper left quadrant, there is a - sign, indicating that the function f maps this quadrant one-to-one and onto the lower half plane, and similarly for quadrants III and IV.

Then there are some specified values of f (at certain points) on the complex plane.

If I had to "find all possible functions f" or "what is f exactly?", how does one apply the Schwarz Reflection Principle in these problems?

If f mapped quadrant I bijectively to the upper half plane, I am guessing that by the Reflection Principle, it maps quadrant II to the lower half plane (as the picture indicates.) Should I treat the imaginary axis, which separates the two quadrants, as the "real axis" over which f gets reflected? If yes, how would that work? Since the analytic continuation of f(z) given in the Reflection Principle is $\bar f(\bar z)$, but conjugation reflects over the real axis only. So how do we get the reflection across the imaginary axis.

Similar question involves looking at two intersecting circles, with symmetric regions, and + / - signs indicating what f does to the sectors of the two circles (and what it does to the outside of the two circles.) How do we handle reflection over arcs ...instead of over the real axis?

Sorry for the wordy question -- I am having lots of trouble understanding the theorem and its application to problems.



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