the Cayley transform
/ KAY-lee /
Complex analysis has two favourite stages on which to act out its theorems: the upper half-plane (everything above the real axis) and the unit disk (everything inside the circle |w| = 1). The half-plane is convenient for some arguments, the disk for others — so it is enormously useful to have one explicit, perfect dictionary translating between them. The Cayley transform is that dictionary: a specific Mobius transformation that maps the upper half-plane conformally and bijectively onto the open unit disk.
Its formula is w = (z - i)/(z + i). Trace what it does. The point z = i (deep inside the upper half-plane) goes to 0 (the center of the disk). The boundary of the half-plane, the real axis, maps to the boundary of the disk, the unit circle: for any real x, the numerator and denominator x - i and x + i have equal modulus (each is the conjugate of the other), so the quotient has modulus exactly 1. The pole at z = -i sits in the LOWER half-plane, safely outside the region we care about, and z = infinity maps to w = 1. Being a Mobius map it is automatically conformal and circle-preserving, so the real-axis-plus-infinity (a 'circle' through infinity) correctly becomes the genuine circle |w| = 1.
Why bother? Because it lets you import every disk result into the half-plane and vice versa, free of charge. The Schwarz lemma, the classification of disk automorphisms, the Poisson integral formula, the hyperbolic metric — all are usually proved on whichever of the two regions is easier, then transported across by the Cayley transform. It is the most-used single conformal map in the whole subject. One caution about orientation: w = (z - i)/(z + i) sends the UPPER half-plane to the disk; the closely related map (z + i)/(z - i) or a sign change sends the LOWER half-plane instead, so check which half you mean before quoting the formula.
Check a boundary point: z = 0 (on the real axis) maps to w = (0 - i)/(0 + i) = -i/i = -1, which lies on the unit circle. And the interior point z = i maps to w = (i - i)/(i + i) = 0, the disk's center — confirming interior goes to interior, boundary to boundary.
The Cayley transform pins i to the disk's center and the real axis to the unit circle.
There is no single 'the' Cayley transform — any Mobius map carrying half-plane to disk earns the name, and they differ by a rotation of the disk. The form (z - i)/(z + i) is just the most common normalization.