Editing Cayley transform (section)

==Complex homography==
[[Image:Cayley transform in complex plane.png|thumb|right| 300px|Cayley transform of upper complex half-plane to unit disk]]
On the [[Upper half-plane|upper half]] of the [[complex plane]], the Cayley transform is:<ref>Robert Everist Green & [[Steven G. Krantz]] (2006) ''Function Theory of One Complex Variable'', page 189, [[Graduate Studies in Mathematics]] #40, [[American Mathematical Society]] {{ISBN|9780821839621}}</ref><ref>[[Erwin Kreyszig]] (1983) ''Advanced Engineering Mathematics'', 5th edition, page 611, Wiley {{ISBN|0471862517}}</ref>
:<math>f(z) = \frac {z - i}{z + i} .</math>
Since <math>\{\infty, 1, -1\}</math> is mapped to <math>\{1, -i, i\}</math>, and [[Möbius transformation]]s permute the [[generalised circle]]s in the [[complex plane]], <math>f</math> maps the real line to the [[unit circle]]. Furthermore, since <math>f</math> is a [[homeomorphism]] and <math>i</math> is taken to 0 by <math>f</math>, the upper half-plane is mapped to the [[unit disk]].

In terms of the [[mathematical model|models]] of [[hyperbolic geometry]], this Cayley transform relates the [[Poincaré half-plane model]] to the [[Poincaré disk model]].

In electrical engineering the Cayley transform has been used to map a [[electrical reactance|reactance]] half-plane to the [[Smith chart]] used for [[impedance matching]] of transmission lines.