Directed infinity

A directed infinity is a type of infinity in the complex plane that has a defined

. 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.

See also

• Point at infinity

References