2021-6.md

course	course_year	question_number	tags	title	year
Complex Analysis	IB	6	IB 2021 Complex Analysis	Paper 4, Section I, $3 G$	2021

Let $f$ be a holomorphic function on a neighbourhood of $a \in \mathbb{C}$. Assume that $f$ has a zero of order $k$ at $a$ with $k \geqslant 1$. Show that there exist $\varepsilon>0$ and $\delta>0$ such that for any $b$ with $0<|b|<\varepsilon$ there are exactly $k$ distinct values of $z \in D(a, \delta)$ with $f(z)=b$.