Cayley transform

Cayley transform maps upper half plane to open unit disk

In complex analysis, the Cayley transform is the map

$\operatorname {W} :z\mapsto {\frac {z-i}{z+i}}.$

The Cayley transform is a linear fractional transformation. It can be extended to an automorphism of the Riemann sphere.

Of particular note are the following facts:

• W maps the real line R injectively into the unit circle T (complex numbers of modulus 1). The image of R is T with 1 removed.

• W maps the upper imaginary axis i [0, ∞) bijectively onto the half-open interval [-1, +1).

• W maps the point at infinity to 1.

• W maps 0 to -1.

• W has a pole at -i (so W maps -i to the point at infinity).

• W maps the upper half plane of C conformally onto the open unit disc of C.

By analogy, the expression Cayley transform is also used to denote a mapping from operators to operators: Aside from questions of domain it associates to a linear operator A the linear operator

$\operatorname {W} (A):(A+i)z\mapsto (A-i)z.$

See self-adjoint operator for details.

Reference

• Walter Rudin, Real and Complex Analysis, McGraw Hill, 1966, ISBN 0-07-100276-6 . (This book is sometimes referred to as Big Rudin or Green Rudin)