Cayley transform

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Cayley transform maps upper half plane to open unit disk
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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 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.

[edit] 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)
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