Cayley transform Guide, Meaning , Facts, Information and Description
In complex analysis, the Cayley transform is the map
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
See
self-adjoint operator for details.
Reference
W. Rudin, Real and Complex Analysis, McGraw Hill, 1966 (This book is commonly referred to as Big Rudin)
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