math121a-f23:hw_3
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math121a-f23:hw_3 [2023/09/10 17:43] pzhou |
math121a-f23:hw_3 [2023/09/13 12:25] (current) pzhou |
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| 3. let $z = 2 e^{i \pi / 3}$. What does $z^i$ mean? is it multivalued? | 3. let $z = 2 e^{i \pi / 3}$. What does $z^i$ mean? is it multivalued? | ||
| - | 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 $z$? | + | 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 $z=0$? | 5. What is the Laurent expansion (first 3 terms) of $\frac{\cos(z)}{z}$ around $z=0$? $\frac{\cos(z)}{\sin(z)}$ around $z=0$? | ||
math121a-f23/hw_3.1694393001.txt.gz · Last modified: 2023/09/10 17:43 by pzhou
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