Quiz+1

1. Sketch the set math \{z\in \mathbf{C}| \left|z^3+1\right|=0\} math

The set consists of three points, -1, e^(pi/3 i) and e^(-pi/3 i)

2. Find the limit math \lim\limits_{n\to \infty} n\left(\frac i 2\right)^n math or explain why it does not exist.

0, because |n (i/2)^n|=n/2^n tends to 0.

3. Sketch the image of the set math \{z| |z|\leq 1, Re {z}\leq Im {z}, Re {z} \geq 0 \} math under the mapping math f(z)=z^4+1+i math

The original set can be written as the set of z=r e^(i t) for 0<=r<=1 and pi/4<=t<=pi/2. After applying z->z^4 the set becomes the set of all numbers of the form r^4 e^(4 i t) with the same inequalities on r and t. Now r^4 can be any number from 0 to 1, while 4t is any number from pi to 2 pi. Hence this set is the bottom semidisc of the unit disc centered at the origin.

After shifting the set one unit up and one unit to the right we get the answer: bottom semidisc of radius 1 centered at 1+i.