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:

• [math]\displaystyle{ z\infty = \sgn(z)\infty \text{ if } z\ne 0 }[/math]
• [math]\displaystyle{ 0\infty\text{ is undefined, as is }\frac{z\infty}{w\infty} }[/math]
• [math]\displaystyle{ a z\infty = \begin{cases} \sgn(z)\infty & \text{if }a \gt 0, \\ -\sgn(z)\infty & \text{if }a \lt 0. \end{cases} }[/math]
• [math]\displaystyle{ w\infty z\infty = \sgn(w z)\infty }[/math]

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

See also

• Point at infinity

References

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