math121a-f23:september_8_friday
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math121a-f23:september_8_friday [2023/09/05 18:18] pzhou created |
math121a-f23:september_8_friday [2026/02/21 14:41] (current) |
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| 2. sin, cos, sinh, cosh (not a big deal, they are linear combination of exp,log) | 2. sin, cos, sinh, cosh (not a big deal, they are linear combination of exp,log) | ||
| - | 3. $\sqrt{z}$? (multivalued function) | + | 3. power and roots. $\sqrt{z}$? (multivalued function) |
| 4. Taylor series, Laurent series. | 4. Taylor series, Laurent series. | ||
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| + | ====== Exercise ====== | ||
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| + | 1. let $z = 2 e^{i \pi / 3}$, | ||
| + | * compute $z^2, z^3$. | ||
| + | * what is $\log z$? (be aware this is a multivalued function) | ||
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| + | 2. how many complex solution does $z^4 = -1$ have? what are they? | ||
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| + | 3. let $z = 2 e^{i \pi / 3}$. What does $z^i$ mean? is it multivalued? | ||
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| + | 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 $z$? | ||
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| + | 5. What is the Laurent expansion (first 3 terms) of $\frac{\cos(z)}{z}$ around $z=0$? $\frac{\cos(z)}{\sin(z)}$ around $z=0$? | ||
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math121a-f23/september_8_friday.1693937880.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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