Cayley Transform - Conformal Map

Conformal Map

In complex analysis, the Cayley transform is a mapping of the complex plane to itself, given by

This is a linear fractional transformation, and can be extended to an automorphism of the Riemann sphere (the complex plane augmented with a point at infinity).

Of particular note are the following facts:

• W maps the upper half plane of C conformally onto the unit disc of C.
• W maps the real line R injectively into the unit circle T (complex numbers of absolute value 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 0 to −1.
• W maps the point at infinity to 1.
• W maps −i to the point at infinity (so W has a pole at −i).
• W maps −1 to i.
• W maps both 1⁄₂(−1 + √3)(−1 + i) and 1⁄₂(1 + √3)(1 − i) to themselves.