Problems to think about when preparing for Test 1

Date: MATH 3160-1 - Fall 2002

Note: These are some interesting problems. They provide additional practice for Test 1 but, beware: they do not necessarily cover all the topics to be evaluated by the test.

1. Prove that $z$ is real or pure imaginary if and only if $(\overline {z})^2=z^2$.

2. For $z_0 = (1-i)(\sqrt{3}+i)$ and $z_1 = \frac{1-i}{\sqrt{3}+i}$
   1. Find the modulus, argument (all possible values), and the principal argument.
   2. Write the exponential (or trigonometric) form.
   3. Sketch a graph showing $z_0$ and $z_1$ on the plane.

3. Find all the fifth roots of $-1-i$ and sketch a graph showing their location.

4. Find the domain of $f(z) = \frac{z^2-2}{2z^3+i}$.

5. Let

   $$U=\{z\in \ensuremath{\mathbb{C}}: {\mathrm{Re}}(z)\geq 2 \vert z\vert \leq 10\}$$

   
   
   and

   $$V=\{z\in \ensuremath{\mathbb{C}}: 1<\vert z\vert<3 \vert{\mathrm{Im}}(z)\vert>2\}$$

   
   
   1. Sketch a graph of $U$ and $V$.
   2. Discuss if each set has the following properties:
      1. Is open.
      2. Is closed.
      3. Is connected.
      4. Is bounded.
      5. Is a domain.

6. Compute the following limits, if they exist. Otherwise give an argument of why they don't exist.
   1. $\lim_{z\rightarrow i}\frac{z^3-i}{z+i}$.
   2. $\lim_{z\rightarrow \infty} \frac{z}{\overline {z}}$.
   3. $\lim_{z\rightarrow 2i}\frac{(z-2i)(z+i)}{z^2+4}$.

7. Check the continuity of

   $$f(z) = \begin{cases}\frac{z-1}{z^2-1}, \text{ if } z\neq \pm 1\\ \frac{1}{2}, \text{ if } z=1\\ 0, \text{ if } z=-1 \end{cases}$$

   
   
   at $z=-1$, $z=0$ and $z=1$. In each case state clearly what you want to check and then prove it.

8. Let

   $$f(x+iy) = \frac{1}{x^2+y^2} ((x^4-y^4+x) + i(2xy(x^2+y^2)-y)).$$

   
   
   Use the Cauchy-Riemann equations (and additional conditions) to determine where $f$ defines an analytic function. Find $f'$ where it is defined.

9. For $f(x+iy)=\arctan(\frac{y}{x}) - i \ln(\sqrt{x^2+y^2})$ let $u={\mathrm{Re}}(f)$.
   1. Use the Cauchy-Riemann equations to verify that $f$ is analytic on $x>0$.
   2. Check explicitly that $u$ is a harmonic function (i.e., compute the second derivatives and check the corresponding equation).
   3. Find $v$, a harmonic conjugate of $u$.
   4. Check that, up to an additive constant, $v = {\mathrm{Im}}(f)$.