What is the meaning of "on each bounded disc of the complex plane"?

Question

My primary language for math is not English.

What is the meaning of "on each bounded disc of the complex plane"?

For example (from https://www.encyclopediaofmath.org/index.php/Uniformly-convergent_series): "... is uniformly convergent on each bounded disc of the complex plane, ..."

More specifically, does "on each bounded disc of the complex plane" refer to the group $A=\{z:|z-z_0|\le r\}$ where $r\in \mathbb{R}$, $z_0\in \mathbb{c}$? (note that $A$ is what I would directly translate from my primary math language to "a closed circle").

Thanks.

Answer 1

Every disc in the complex plane (open or closed) is a bounded set. A closed disc is a compact set.

Suppose that $R$ is a fixed positive number, and that $z_0$ is a fixed complex number.

• The circle of radius $R$ centred at $z_0$ is defined by $C(z_0; R) := \{z \in \mathbb{C} : |z − z_0| = R\}$
• The open disc of radius $R$ centred at $z_0$ is defined by $D(z_0; R) := \{z \in \mathbb{C} : |z − z_0| < R\}$
• The closed disc of radius $R$ centred at $z_0$ is defined by $D(z_0; R) := \{z \in \mathbb{C} : |z − z_0| \leq R\} = D(z_0; R) ∪ C(z_0; R)$

And finally,

• A subset $U$ of $\mathbb{C}$ is said to be bounded if there is some positive number $\Delta$ such that $U \subseteq D(0; ∆)$, i.e., $|z| \leq \Delta$ for every $z \in U$. $U$ is unbounded if it is not bounded.