| course |
course_year |
question_number |
tags |
title |
year |
Complex Analysis |
IB |
8 |
|
Paper 3, Section II, G |
2014 |
State the Residue Theorem precisely.
Let $D$ be a star-domain, and let $\gamma$ be a closed path in $D$. Suppose that $f$ is a holomorphic function on $D$, having no zeros on $\gamma$. Let $N$ be the number of zeros of $f$ inside $\gamma$, counted with multiplicity (i.e. order of zero and winding number). Show that
$$N=\frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z$$
[The Residue Theorem may be used without proof.]
Now suppose that $g$ is another holomorphic function on $D$, also having no zeros on $\gamma$ and with $|g(z)|<|f(z)|$ on $\gamma$. Explain why, for any $0 \leqslant t \leqslant 1$, the expression
$$I(t)=\int_{\gamma} \frac{f^{\prime}(z)+\operatorname{tg}^{\prime}(z)}{f(z)+\operatorname{tg}(z)} d z$$
is well-defined. By considering the behaviour of the function $I(t)$ as $t$ varies, deduce Rouché's Theorem.
For each $n$, let $p_{n}$ be the polynomial $\sum_{k=0}^{n} \frac{z^{k}}{k !}$. Show that, as $n$ tends to infinity, the smallest modulus of the roots of $p_{n}$ also tends to infinity.
[You may assume any results on convergence of power series, provided that they are stated clearly.]