2016-2017第一学期复变函数期末考试

一、求\(\displaystyle \int_{C}\dfrac{2z^2-z+1}{(z-1)(z-3)}\),其中\(C:|z|=2\).

二、\(f(w)\)\(\{w|1<|w|<+\infty\}\)内解析,证明:\(\displaystyle \int_{C}f(\dfrac{1}{z^2})\mathrm{d}z=0\).

三、将\(\dfrac{1}{z(z-1)}\)\(\infty\)处洛朗展开,并判断奇点类型\((1<|z|<+\infty)\).

四、证明:\(f(z)\)\(z_0\)\(n\)阶极点\(\Longleftrightarrow\)存在\(\varphi(z)=(z-z_0)^nf(z)\).

五、

(1)写出\(f(z)\)\(f'(z)\)在邻域\(B_2(0)\)的柯西积分公式.

(2)已知\(|f'(z)|\leqslant 2M\),若\(f(z)\leqslant M\)\(\forall z \in \partial B_2(0)\),证明\(f'(z)\leqslant 2M\)\(\forall z \in B_1(0)\).

(3)\(f'(z)\)\(B_2(0)\)有界还是无界?

六、已知\(f(z)\)的零点和极点都在邻域\(B_1(0)\)中.

(1)写出\(f(z)\)\(B_1(0)\)上的幅角原理.

(2)证明\(N(f+g,B_1(0))-P(f+g,B_1(0))=N(f,B_1(0))-P(f,B_1(0))\).

(3)\(\sqrt[n]{f(z)}\)不以\(\infty\)为支点,则\(n\)\(N(f,B_1(0))-P(f,B_1(0))\)有什么关系?


Last update: July 26, 2020