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Homework 2 (MATH 1199 Fall 2019)
1. In the following problems, write in polar form and multiply $z_1$ and $z_2$, and then express that product in the form $a+bi$, where $a, b \in \mathbb{R}$.
(a) $z_1=2+2i$ and $z_2=3i$
(b) $z_1=-\sqrt{3}-i$ and $z_2=5$
(c) $z_1=1-i$ and $z_2=5+5i$
(d) $z_1=3i$ and $z_2=-4i$
2. Find $\mathrm{Arg} \left( \dfrac{2i}{1+i} \right)$ in any way you like.
3. Compute by hand and express the final answer in the form $a+bi$ for $a,b \in \mathbb{R}$.
(a) $(-1+\sqrt{3}i)^{14}$
(b) $(-2-2\sqrt{3}i)^{11}$
(c) $(4-4i)^{-11}$
4. Find all fifth roots of $1$.
5. Find all sixth roots of $1$.
6. Find all cube roots of $i$ and write them in the form $a+bi$ for $a,b \in \mathbb{R}$.
7. Find all fourth roots of $1+i$.
8. Find all cube roots of $-\sqrt{3}-i$.
9. Find all fourth roots of $-1$ and write them in the form $a+bi$ for $a,b \in \mathbb{R}$.
10. Write the described function in the form $u+iv$ where $u, v \colon \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$.
(a) $\left\{ \begin{array}{ll} f \colon \mathbb{C} \rightarrow \mathbb{C} \\ f(z)=z^2-2\overline{z} \end{array} \right.$
(b) $\left\{ \begin{array}{ll} g \colon \mathbb{C} \setminus \{-1\} \rightarrow \mathbb{C} \\
g(z)=\dfrac{z}{z+1}
\end{array} \right.$
(c) $\left\{ \begin{array}{ll} h \colon \mathbb{C} \setminus \{1+i\} \rightarrow \mathbb{C} \\
h(z) = \dfrac{z^2}{z-1-i} \end{array} \right.$
(d) $\left\{ \begin{array}{ll} \ell \colon \mathbb{C} \setminus \{0\} \rightarrow \mathbb{C} \\
\ell(z) = \dfrac{\overline{z}}{|z|} \end{array} \right.$
