Cayley transform
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In complex analysis, the Cayley transform is the map
- <math> \operatorname{W}:z \mapsto \frac{z-i}{z+i}. </math>
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
- <math> \operatorname{W}(A): (A + i)z \mapsto (A - i) z .</math>
See self-adjoint operator for details.
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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)