## `complexInfinity` -- complex infinity

#### Introduction

`complexInfinity` represents the only non-complex point of the one-point compactification of the complex numbers.

#### Related Functions

`infinity`

#### Details

- Mathematically, `complexInfinity` is the north pole of the Riemann sphere, with the unit circle as equator and the point 0 at the south pole.
- With respect to arithmetic, `complexInfinity` behaves like ```1/0`''. In particular, nonzero complex numbers may be multiplied or divided by `complexInfinity` or `1/`complexInfinity. Adding `complexInfinity` to a nonzero number yields `undefined`.
- With respect to arithmetical operations, `complexInfinity` is incompatible with the real `infinity`.

#### Example 1

`complexInfinity` can be used in arithmetical operations with complex numbers. The result in multiplications or divisions is either `complexInfinity`, `0`, or `undefined`:

    >> 3*complexInfinity, I*complexInfinity, 0*complexInfinity;
       3/complexInfinity, I/complexInfinity, 0/complexInfinity;
       complexInfinity/3, complexInfinity/I;
       complexInfinity*complexInfinity, complexInfinity/complexInfinity;

                    complexInfinity, complexInfinity, undefined

                                      0, 0, 0

                         complexInfinity, complexInfinity

                            complexInfinity, undefined

Symbolic expressions in multiplications or divisions involving `complexInfinity` are implicitly assumed to be different from both `0` and `complexInfinity`:

    >> delete x:
       x*complexInfinity, x/complexInfinity, complexInfinity/x

                        complexInfinity, 0, complexInfinity

The result in additions is always `undefined`:

    >> 3 + complexInfinity, I + complexInfinity, x + complexInfinity

                          undefined, undefined, undefined

#### Background

- `complexInfinity` is the only element of the domain `stdlib::CInfinity`.

#### Changes

- no changes

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