Mapping Disks

Complex Analysis · Axiom Academy

The Cayley map turns the upper half-plane into the unit disk, and Blaschke factors slide any point of the disk to its center — angles intact. 1. The Cayley Map: Half-Plane to Disk The upper half-plane stretches off to infinity, which makes it awkward to draw or compute on. The Cayley map folds all of it into the unit disk with a single fraction. Watch the shaded half-plane and its boundary — the real axis — sweep inside the circle. Sends the upper half-plane onto the open disk and the real axis onto the unit circle 2. Blaschke Factors: Sliding the Disk Onto Itself Now keep the disk fixed and ask: which maps send onto itself , one-to-one? The answer is the Blaschke factors . Pick any interior point a , and glides it to the center while every boundary point stays exactly on the circle. Watch a slide to 0 as the whole disk flows with it. — whichever point you choose lands at the origin. If |z| = 1 then : the unit circle maps onto itself. With , — the center is pushed out toward -a . |a| and set where to move the origin; adds a spin. Worked instance: send to the origin 3. Why It All Works: Angles Are Preserved Both families share one property that makes them so powerful: they are conformal — they preserve angles. Draw a tiny right-angle cross anywhere in the source and its image is still a right-angle cross. Watch the perpendicular cross ride the Cayley map into the disk and stay perpendicular.

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