Is there an analog of $\epsilon$ in complex analysis?

Question

When someone writes let $\epsilon > 0$ in the context of calculus, we know the intention is that $\epsilon$ is real, and may be made as small as we like. Is there a corresponding symbol (Greek letter or otherwise) that indicates a similar idea in the complex realm? In other words, a complex number whose magnitude is arbitrarily small?

Answer 1

When you make this kind of set up, you want to deal with the numbers within a bounded set, say an interval $(-\epsilon, \epsilon)$.

In general, you can use a ball in $\mathbb{R}^n$ of radius $\epsilon$. Then you can just use that $\mathbb{C} = \mathbb{R}^2$ and do the same.

On $\mathbb{C}$ you can use the disk $\{z ; |z| < \epsilon\}$. In higher dimensions it is often more useful to use products of discs, called polydiscs, instead of balls.