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.

Sequences

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 $z\in \mathbb{C}$.

Series

15. Does the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{i}^n}$ converge?
16. To what does the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{2^n}{(3+i{)}^n}}$ converge?
17. Does the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}}$ converge?
18. Prove that ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{z}_n}$ converges if and only the series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\mathsf{Re}(z_n)}$ and ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\mathsf{Im}(z_n)}$ both converge.