(Due Wednesday, Sep 13)

0. Read Boas Ch2, section 1 - 9, find 5 interesting problems there and do it. (copy down the problem, so the grader / reader know which one you are doing).

1. let $z = 2 e^{i \pi / 3}$,

• compute $z^2, z^3$.
• what is $\log z$? (be aware this is a multivalued function)

2. how many complex solution does $z^4 = -1$ have? what are they?

3. let $z = 2 e^{i \pi / 3}$. What does $z^i$ mean? is it multivalued? How about $z^{1/2}$?

4. express $\sin(1+2 i)$ in terms of exponential. Is it true that $\sin(z) = Im( e^{i z})$ for all real $z$, for all complex $z$? (corrected, previous question was asking $\sin(z) = Re( e^{i z})$, which is false even for $z$ real)

5. What is the Laurent expansion (first 3 terms) of $\frac{\cos(z)}{z}$ around $z=0$? $\frac{\cos(z)}{\sin(z)}$ around $z=0$?