| course |
course_year |
question_number |
tags |
title |
year |
Complex Analysis or Complex Methods |
IB |
9 |
IB |
2021 |
Complex Analysis or Complex Methods |
|
Paper 1, Section II, G |
2021 |
(a) State a theorem establishing Laurent series of analytic functions on suitable domains. Give a formula for the $n^{\text {th }}$ Laurent coefficient.
Define the notion of isolated singularity. State the classification of an isolated singularity in terms of Laurent coefficients.
Compute the Laurent series of
$$f(z)=\frac{1}{z(z-1)}$$
on the annuli $A_{1}={z: 0<|z|<1}$ and $A_{2}={z: 1<|z|}$. Using this example, comment on the statement that Laurent coefficients are unique. Classify the singularity of $f$ at 0 .
(b) Let $U$ be an open subset of the complex plane, let $a \in U$ and let $U^{\prime}=U \backslash{a}$. Assume that $f$ is an analytic function on $U^{\prime}$ with $|f(z)| \rightarrow \infty$ as $z \rightarrow a$. By considering the Laurent series of $g(z)=\frac{1}{f(z)}$ at $a$, classify the singularity of $f$ at $a$ in terms of the Laurent coefficients. [You may assume that a continuous function on $U$ that is analytic on $U^{\prime}$ is analytic on $U$.]
Now let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function with $|f(z)| \rightarrow \infty$ as $z \rightarrow \infty$. By considering Laurent series at 0 of $f(z)$ and of $h(z)=f\left(\frac{1}{z}\right)$, show that $f$ is a polynomial.
(c) Classify, giving reasons, the singularity at the origin of each of the following functions and in each case compute the residue:
$$g(z)=\frac{\exp (z)-1}{z \log (z+1)} \quad \text { and } \quad h(z)=\sin (z) \sin (1 / z)$$