Directed infinity

A directed infinity is a type of infinity in the complex plane that has a defined complex argument θ but an infinite absolute value r. For example, the limit of 1/x where x is a positive real number approaching zero is a directed infinity with argument 0; however, 1/0 is not a directed infinity, but a complex infinity. Some rules for manipulation of directed infinities (with all variables finite) are:


 * $$z\infty = \sgn(z)\infty \text{ if } z\ne 0$$
 * $$0\infty\text{ is undefined, as is }\frac{z\infty}{w\infty}$$
 * $$a z\infty = \begin{cases} \sgn(z)\infty & \text{if }a > 0, \\ -\sgn(z)\infty & \text{if }a < 0. \end{cases} $$
 * $$w\infty z\infty = \sgn(w z)\infty$$

Here, sgn(z) = $z⁄|z|$ is the complex signum function.