Week 1 Worksheet

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Complex Numbers

1. Find the modulus, all arguments and the principal value of the argument for the following complex numbers.
   1. $2i$
   2. $1-i\sqrt{3}$
   3. $-4$
2. Draw the set of all $z\in \mathbb{C}$ satisfying the following conditions.
   1. $\mathsf{Re}(z)>2$
   2. $1<\mathsf{Im}(z)<2$
   3. $|z|<3$
   4. $|z-2|<|z+1|$

Arithmetic

3. What is $2+3i$ multiplied by $-5+i$?
4. Write the following expressions in the form $x+iy$, $x,y\in \mathbb{R}$.
   1. $(3+4i{)}^2\phantom{{\displaystyle \frac{1}{1}}}$
   2. ${\displaystyle \frac{2+3i}{3-4i}}$
   3. ${\displaystyle \frac{1-5i}{3i-1}}$
   4. ${\displaystyle \frac{1-i}{1+i}}-i+2$
   5. ${\displaystyle \frac{1}{i}}$
5. Find all solutions of the following equations.
   1. $z^2=-5+12i$
   2. $z^2+4z+12-6i=0$
6. Let $z,w\in \mathbb{C}$. Prove the following statements.
   1. $\mathsf{Re}(z\pm w)=\mathsf{Re}(z)\pm \mathsf{Re}(w)$
   2. $\mathsf{Im}(z\pm w)=\mathsf{Im}(z)\pm \mathsf{Im}(w)$
7. Give examples to show that neither $\mathsf{Re}(zw)=\mathsf{Re}(z)\mathsf{Re}(w)$ nor $\mathsf{Im}(zw)=\mathsf{Im}(z)\mathsf{Im}(w)$ hold in general.
8. Fix $z,w\in \mathbb{C}$. Prove the following statements.
   1. $\overline{z\pm w}=\overline{z}\pm \overline{w}$
   2. $\overline{zw}=\overline{z}\overline{w}$
   3. $\overline{1/z}=1/\bar{z}$ if $z\ne 0$
   4. $z+\overline{z}=2\mathsf{Re}(z)$
   5. $z-\overline{z}=2i\mathsf{Im}(z)$
9. Recall that every non-zero $z\in \mathbb{C}$ can be written in polar form as $r(\cos \theta +i\sin \theta )$.
   1. Use induction on $n$ to derive $$(\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta$$ for all $n\in \mathbb{N}$. (This is called de Moivre's Theorem.)
   2. Use De Moivre's Theorem to derive formulae for $\cos (3\theta )$, $\sin (3\theta )$, $\cos (4\theta )$, $\sin (4\theta )$ in terms of $\cos \theta$ and $\sin \theta$.
10. Let $w_0\ne 0$ be a complex number such that $|w_0|=r$ and $\mathsf{Arg}(w_0)=\theta$. Find the polar forms of all the solutions $z$ to $z^n=w_0$, where $n\ge 1$ is a positive integer.
11. Let $\mathsf{Arg}(z)$ denote the principal value of the argument of $z$. Give an example to show that $$\mathsf{Arg}(z_1{z}_2)\ne \mathsf{Arg}(z_1)+\mathsf{Arg}(z_2)$$ in general.

Distance and Limits

12. A sequence $z_n$ in $\mathbb{C}$ is known to converge to $a$. Use the reverse triangle inequality to prove that $|z_n|$ converges to $|a|$.
13. Determine whether the following sequences converge.
    1. $z_n=(1+i{)}^n\phantom{{\displaystyle \frac{1}{1}}}$
    2. $z_n={\displaystyle \frac{(1+i{)}^n}{n}}$
    3. $z_n={\displaystyle \frac{1}{(1+i{)}^n}}$
14. Prove that $z^n/n!$ converges to zero for every complex number $z\in \mathbb{C}$.

Paths and Domains

15. Verify that $\gamma (t)=4t^2+2it$ on $[1,2]$ is continuous.
16. Which of the following could be the image of a path?
    1. 
    2. 
    3. 
    4. 
17. Write down a formula for a path $\gamma :[0,1]\to \mathbb{C}$ describing a path from $2$ to $i-1$.
18. Sketch the following sets of complex numbers. Which of them are domains?
    1. $\{z\in \mathbb{C}:\mathsf{Im}(z)>0\}$
    2. $\{z\in \mathbb{C}:\mathsf{Re}(z)>0\text{ and }|z|<2\}$
    3. $\{z\in \mathbb{C}:|z-2|<1\text{ or }|z+2|<1\}$
    4. $\mathbb{C}\setminus \{z\in \mathbb{C}:x\le 0\text{ and }y=0\}$
19. Let $E$ and $F$ be open subsets of $\mathbb{C}$. Prove that $E\cap F$ is open.
20. Is the intersection of two domains always a domain? Either prove this or provide a counterexample.