---
tag: 0X45ZM731G
kind: example
name: Cayley transform
aliases: [cayley-transform-of-the-upper-half-plane]
msc: [30C20]
instance_of: [0SG74WBFA2]
uses: [0B346DDAM5]
sources:
  - url: https://en.wikipedia.org/wiki/Cayley_transform
    locator: Complex homography
---

## Example

Let $\mathbb{H} = \{ z \in \mathbb{C} : \operatorname{Im} z > 0 \}$ be the upper half-plane and $\mathbb{D} = \{ w \in \mathbb{C} : |w| < 1 \}$ the open unit disc.
The Cayley transform is
$$W(z) = \frac{z - i}{z + i}.$$
It is a [[Möbius transformation|0B346DDAM5]] with $a = 1$, $b = -i$, $c = 1$, $d = i$, and $ad - bc = 2i \neq 0$.
For $z \in \mathbb{C}$, $|W(z)| < 1$ holds exactly when $z$ is nearer to $i$ than to $-i$, that is, when $\operatorname{Im} z > 0$.
The map $W$ is a bijection from $\mathbb{H}$ onto $\mathbb{D}$, with inverse $w \mapsto i \frac{1 + w}{1 - w}$.

## Remarks

Some values are $W(i) = 0$, $W(0) = -1$, $W(1) = -i$, $W(-1) = i$ and $W(\infty) = 1$.
The map $W$ sends the real line together with $\infty$ onto the unit circle.
It is an explicit case of the [[Riemann mapping theorem|02WHMEG5ZR]] for $\mathbb{H}$.
In hyperbolic geometry, it connects the half-plane model and the disc model.

## Explanations

### L1 human

The Cayley transform turns the upper half-plane into the unit disc.
A point of $\mathbb{H}$ is nearer to $i$ than to $-i$, so the quotient of the two distances is less than $1$.
It lets you move a problem on the half-plane to the disc, where it can be easier.