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:

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

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