Extended complex plane

The complex plane together with the point at infinity, written $\mathbb{C}\cup\{\infty\}$ .

The extended complex plane, written $\mathbb{C}\cup\{\infty\}$ or $\widehat{\mathbb{C}}$ , is the ordinary complex plane together with the single point at infinity adjoined to it. Working in the extended plane lets statements about limits and behavior 'as $z\to\infty$ ' be phrased as ordinary limits at an ordinary point, rather than as a separate special case.

Examples

In the extended complex plane, $f(z)=1/z$ can be regarded as continuous at $z=0$ once $f(0)$ is defined to be $\infty$ .
A polynomial of degree $n\ge 1$ sends $\infty$ to $\infty$ in the extended complex plane.