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--- math121a-f23:september_8_friday [2023/09/08 14:11]
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+++ math121a-f23:september_8_friday [2026/02/21 14:41] (current)
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 . Taylor series, Laurent series.
+====== Exercise ======
+. let $z = 2 e^{i \pi / 3}$,
+  * compute $z^2, z^3$.
+  * what is $\log z$? (be aware this is a multivalued function)
+. how many complex solution does $z^4 = -1$ have? what are they?
+. let $z = 2 e^{i \pi / 3}$. What does $z^i$ mean? is it multivalued? How about $z^{1/2}$?
+. 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 $z$?
+. What is the Laurent expansion (first 3 terms) of $\frac{\cos(z)}{z}$ around $z=0$? $\frac{\cos(z)}{\sin(z)}$ around $z=0$?

Lecture Notes