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.^[1] 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:

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

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

• Point at infinity

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