| course |
course_year |
question_number |
tags |
title |
year |
Complex Analysis or Complex Methods |
IB |
11 |
IB |
2010 |
Complex Analysis or Complex Methods |
|
Paper 2, Section II, A |
2010 |
(a) Prove that a complex differentiable map, $f(z)$, is conformal, i.e. preserves angles, provided a certain condition holds on the first complex derivative of $f(z)$.
(b) Let $D$ be the region
$$D:={z \in \mathbb{C}:|z-1|>1 \text { and }|z-2|<2}$$
Draw the region $D$. It might help to consider the two sets
$$\begin{aligned}
&C(1):={z \in \mathbb{C}:|z-1|=1} \\
&C(2):={z \in \mathbb{C}:|z-2|=2}
\end{aligned}$$
(c) For the transformations below identify the images of $D$.
Step 1: The first map is $f_{1}(z)=\frac{z-1}{z}$,
Step 2: The second map is the composite $f_{2} f_{1}$ where $f_{2}(z)=\left(z-\frac{1}{2}\right) i$,
Step 3: The third map is the composite $f_{3} f_{2} f_{1}$ where $f_{3}(z)=e^{2 \pi z}$.
(d) Write down the inverse map to the composite $f_{3} f_{2} f_{1}$, explaining any choices of branch.
[The composite $f_{2} f_{1}$ means $f_{2}\left(f_{1}(z)\right)$.]