| course |
course_year |
question_number |
tags |
title |
year |
Complex Analysis or Complex Methods |
IB |
10 |
IB |
2021 |
Complex Analysis or Complex Methods |
|
Paper 2, Section II, B |
2021 |
(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function and let $a>0, b>0$ be constants. Show that if
$$|f(z)| \leqslant a|z|^{n / 2}+b$$
for all $z \in \mathbb{C}$, where $n$ is a positive odd integer, then $f$ must be a polynomial with degree not exceeding $\lfloor n / 2\rfloor$ (closest integer part rounding down).
Does there exist a function $f$, analytic in $\mathbb{C} \backslash{0}$, such that $|f(z)| \geqslant 1 / \sqrt{|z|}$ for all nonzero $z ?$ Justify your answer.
(b) State Liouville's Theorem and use it to show the following.
(i) If $u$ is a positive harmonic function on $\mathbb{R}^{2}$, then $u$ is a constant function.
(ii) Let $L={z \mid z=a x+b, x \in \mathbb{R}}$ be a line in $\mathbb{C}$ where $a, b \in \mathbb{C}, a \neq 0$. If $f: \mathbb{C} \rightarrow \mathbb{C}$ is an entire function such that $f(\mathbb{C}) \cap L=\emptyset$, then $f$ is a constant function.