Today we are going to meet with some old friends, which will serve as anchor when we go out and meet with more exotic ones.

1. exponential and log

2. sin, cos, sinh, cosh (not a big deal, they are linear combination of exp,log)

3. power and roots. $\sqrt{z}$? (multivalued function)

4. Taylor series, Laurent series.

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 $\sin(z) = Re( e^{i z})$ for all real $z$, for all complex $z$?

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