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# `complexInfinity`

Complex infinity

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## Syntax

```complexInfinity
```

## Description

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

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, non-zero complex numbers may be multiplied or divided by `complexInfinity` or ```1/ complexInfinity```. Adding `complexInfinity` to a finite number yields again `complexInfinity`.

With respect to arithmetical operations, `complexInfinity` is incompatible with the real `infinity`.

## Examples

### 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;```

The result in additions is `undefined` if one of the operands is infinite, and `complexInfinity` otherwise:

```complexInfinity + complexInfinity, infinity + complexInfinity; 3 + complexInfinity, I + complexInfinity, PI + complexInfinity```

Symbolic expressions in arithmetical operations involving `complexInfinity` are implicitly assumed to be different from both `0` and `complexInfinity`:

```delete x: x*complexInfinity, x/complexInfinity, complexInfinity/x, x + complexInfinity```

## Algorithms

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