Mapping Half-Planes

Complex Analysis · Axiom Academy

One Möbius map turns the upper half-plane into a disk — and that single dictionary lets you move any problem onto whichever region is easier. 1. The Cayley Transform Folds ℍ onto 𝔻 Take the boundary of the half-plane — the whole real axis — and watch where the Cayley transform sends it. Every real point lands on the unit circle, and the half-plane above it gets folded neatly into the disk inside. carries onto , the real axis onto |w|=1 2. The Maps That Keep ℍ to Itself Which Möbius maps send the half-plane back onto the half-plane ? The answer is clean: exactly those with real coefficients and a positive determinant. Real coefficients pin the real axis to itself; the sign of ad-bc decides whether "above" stays above. Watch a point in the upper half-plane move as the map is applied. It slides to a new spot — but it never crosses the real axis. The boundary just shuffles along itself. has the same sign as — the upper half-plane maps onto itself. This is a genuine automorphism of . The sign flips: is sent to the lower half-plane instead. Same formula, wrong half. 3. Sending Any Point to the Center The Cayley transform centers the special point i . But you can center any point z_0 you like in with one tweak: subtract z_0 on top and its conjugate on the bottom. The numerator vanishes exactly at z_0 , so z_0 lands at the origin. The denominator's only zero is — and since z_0 is in the upper half-plane, is in the lower half-plane, where it can never spoil the map.

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