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# 复变函数 ygw 2021 Fall B卷 | ||
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一、填空题: | ||
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1. $f(z) = \ln(1+z^3)$ 在 $z=0$ 处的 2022 阶导数为:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_。 | ||
2. $\sum\limits_{n=0}^{+\infty} \frac{1}{(2n+1)!z^{4n}}$ 的和函数为:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_。 | ||
3. $f(z)$ 为整函数,在 $\C$ 上有 $|f(z)| \leq |z|^2$ ,$[f(i)]^2 + \sqrt 2 f(i) + \frac{1}{4} = 0$ , $f(z) =$ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_。 | ||
4. $\sum\limits_{n=0}^{+\infty} (\frac{(4n)^n}{n!})^3z^{2n}$ 的收敛半径为 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_。 | ||
5. $f(z) = \sin(|z|^2+2iz+\bar z^2)$ ,求 $\left.\frac{\partial f}{\partial z}\right|_{z=i\pi}=$ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_。 | ||
6. $\sum\limits_{n=1}^{+\infty}(\frac{1}{\sqrt n} \sinh\frac{i}{\sqrt n}) z^n$ 在 $z = e^{\frac{i\pi}{6}}$ 处的敛散性为 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_。 | ||
7. $C$ 为椭圆 $x^2+4y^2=9$ 的正向曲线,$\oint_C \bar z\ \text dz$ = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_。 | ||
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二、用留数计算定积分: | ||
$$ | ||
\int_{-\infty}^{+\infty} \frac{x\sin x}{x^2-2x+10} \text dx | ||
$$ | ||
三、考虑 $f(z) = z \cos\frac{z}{z-1} + \frac{1}{z}-\frac{1}{\sin z}$ | ||
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1. 给出 $f$ 在扩充复平面 $\overline{\C}$ 上所有奇点的类型,并说明原因 | ||
2. 令 $f(z)$ 在 $0 < |z-1| < 1$ 上的 Laurent 展开式为 $f(z) = \sum\limits_{n=0}^{+\infty} c_n(z-1)^n$ ,求 $c_{-2021}$ | ||
3. 在 $f(z)$ 所有非可去奇点的孤立奇点处求留数 | ||
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四、$f(z) = u(x,y) + i v(x,y)$ 是整函数, $v(x,y) = e^{2\pi x} \sin(Ay) + Bx^2+Cxy - 2y^2$ ,其中 $A,B,C \in \R$ 且 $A \leq 0$,$f(1+i) = i$ ,求 $f(z)$ | ||
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五、$f(z)$ 是整函数,令 $f'(z) = A(x,y) + iB(x,y)$ ,令 $K = \{f(z)\ 是整函数 \mid f'(z)\ 满足 \ A^2+4A \leq 0, \forall (x,y) \in \R^2\}$ | ||
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1. 写出 $K$ 集合 | ||
2. 证明你的结论。 |
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