Week 2 Worksheet

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Paths

1. Verify that $\gamma (t)=4t^2+2it$ on $[1,2]$ is continuous.
2. Which of the following could be the image of a path?
   1. 
   2. 
   3. 
   4. 
3. Write down a formula for a path $\gamma :[0,1]\to \mathbb{C}$ describing a path from $2$ to $i-1$.

Domains

4. 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. $\mathbb{C}\setminus \{z\in \mathbb{C}:x\le 0\text{ and }y=0\}$
5. Let $E$ and $F$ be open subsets of $\mathbb{C}$. Prove that $E\cap F$ is open.
6. Is the intersection of two domains always a domain? Either prove this or provide a counterexample.

Functions

7. What are the real and imaginary parts of $f(z)=\mathsf{Re}(z)$ and $f(z)=|z|$?
8. Let $f(z)=z^2$ on $\mathbb{C}$.
   1. Determine the real and imaginary parts of $f$.
   2. Calculate the gradients of $u$ and $v$. (Recall that $$(\mathrm{\nabla }h)(x,y)=⟨({\partial }_{1}h)(x,y),({\partial }_{2}h)(x,y)⟩$$ is the gradient of $h:{\mathbb{R}}^2\to \mathbb{R}$.)
   3. Verify that $\mathrm{\nabla }u$ and $\mathrm{\nabla }v$ are perpendicular.
9. Let $f(z)=1/z$ on $\mathbb{C}\setminus \{0\}$.
   1. What are the real and imaginary parts of $f$?
   2. What are the level curves of the real part?

Continuous Functions

10. Prove that $f(z)=1/z$ is continuous on $\mathbb{C}\setminus \{0\}$.
11. For each of the following, determine the limit or explain why it does not exist.
    1. $\underset{z\to 0}{lim}{\displaystyle \frac{|z|}{z}}$
    2. $\underset{z\to 0}{lim}{\displaystyle \frac{|z{|}^2}{z}}$
    3. $\underset{z\to 0}{lim}\mathsf{Arg}(z)$
12. Prove that the following functions are continuous.
    1. $f(z)=\mathsf{Re}(z)$ on $\mathbb{C}$
    2. $f(z)=|z|$ on $\mathbb{C}$
    3. $f(z)=\mathsf{Arg}(z)$ on $\mathbb{C}\setminus (-\mathrm{\infty },0]$