MathDB

For each non-zero complex number

z,

let

\arg(z)

be the unique real number

t

such that \minus{}\pi < t \leq \pi and z \equal{} |z|(\cos(t) \plus{} \textrm{i} sin(t)). Given a real number

c > 0

and a complex number

z \neq 0

with

\arg z \neq \pi,

define B(c, z) \equal{} \{b \in \mathbb{R} \ ; \ |w \minus{} z| < b \Rightarrow |\arg(w) \minus{} \arg(z)| < c\}. Determine necessary and sufficient conditions, in terms of

c

and

z,

such that

B(c, z)

has a maximum element, and determine what this maximum element is in this case.

Problem Statement