Tutorial 1 (Week 2)

Not all examples should be covered.

Section 1.1

1. Give the complex number whose polar coordinates $(r, \theta)$ are $(3, \pi/3)$.

2. Show that $\cos (nt)$ can be expressed as a combination of powers of $\cos (t)$ with integer coefficients. (Hint: Use De Moivre’s Theorem and the fact that $\sin^2(t)=1-\cos^2(t)$.

Section 1.1.1

1. Prove that $ z_1z_2=0 \implies (z_1=0) \vee (z_2=0)$ (where $\vee$ means or).
2. Prove that $ z^2=-1 \implies z=\pm i$.

Section 1.2

1. Describe the locus of points $z$ satisfying the given equation: \begin{align} \{z\colon |z+1| +|z-1|=3\};\\ \{z\colon |z+1|-|z-1|=1\}. \end{align}
2. Find all solutions of $z^3=1+i$.

Section 1.3

Domain means connected open set.

1. Describe set $A = \{z = x + iy\colon (x\ge 1) \wedge (y \le -1)\}$ where $\wedge$ means and.

2. Describe the set of points $z^2$ as $z$ varies over the second quadrant: $\{z = x + iy\colon (x < 0) \wedge (y > 0)\}$. Show that this is an open, connected set. (Hint: Use the polar representation of $z$.)

3. An open set $\mathcal{D}$ is star-shaped if there is some point $p$ in $\mathcal{D}$ with the property that the line segment from $p$ to $z$ lies in $\mathcal{D}$ for each $z$ in $\mathcal{D}$.

   1. Show that the disc $\{z: |z — z_0 | < r\}$ is star-shaped,
   2. Show that any convex set is starshaped.
   3. Example of star-shaped but not convex set.