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:

• $z\mathrm{\infty }=\mathrm{sgn}(z)\mathrm{\infty }\text{ if }z\ne 0$
• $0\mathrm{\infty }\text{ is undefined, as is }\frac{z\mathrm{\infty }}{w\mathrm{\infty }}$
• $az\mathrm{\infty }=\{\begin{array}{ll}\mathrm{sgn}(z)\mathrm{\infty }& \text{if }a>0,\\ -\mathrm{sgn}(z)\mathrm{\infty }& \text{if }a<0.\end{array}$
• $w\mathrm{\infty }z\mathrm{\infty }=\mathrm{sgn}(wz)\mathrm{\infty }$

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

See also

• Point at infinity