# 复分析作业

Eufisky posted @ 2017年3月15日 04:22 in 复分析 with tags 复分析 , 1302 阅读

1.设$\displaystyle f_n\in O(D),f_n\rightrightarrows f$, $\displaystyle S_n=\cup_{k\geq n}f_k(D)$,令$\displaystyle S=\cap_{n=1}^\infty S_n$.问$S$的内部$\mathring{S}$和$f(D)$有什么关系?

2.Schwarz引理: $f:\triangle\to \triangle$解析, $f(0)=0$,则$|f'(0)|\leq 1$,并且$|f'(0)|= 1\Leftrightarrow f\in \mathrm{Aut} (\triangle)$.

(1)求出$\lambda$;

(2)若$f\in \mathscr{F}_{z_0}$,则$f\in \mathrm{Aut} (\triangle) \Leftrightarrow |f'(z_0)|=\lambda$.

3.设$f_n,g_n\in O(\triangle),n\geq 1$,设$|f_n(z)|\leq 2,|g_n(z)|\leq 2,|f_n(z)g_n(z)|\leq\frac1n,\forall z,\forall n$.设$f_n(0)=0,g_n(0)=1,\forall n$.设$\{z_n\}$是$\triangle$中子列,若满足$f_n(z_n)=1,g_n(z_n)=0,\forall n$,则$\lim\limits_{n\to\infty}|z_n|=1$.

4.对$r\in (0,1)$,记$A_r=\{z\in \mathbb{C}|r<|z|<1\}$

(1)对$\forall r\in (0,1)$,证明$A_r,\triangle^\ast,\mathbb{C}^\ast$不共形等价;

(2)设$r_1,r_2\in (0,1)$,若$A_{r_1}\cong A_{r_2}$,则$r_1=r_2$;

(3)计算$A_r$的自同构群$\mathrm{Aut}(A_r)$.

5.证明$\mathbb{C}^\ast,\triangle^\ast,A_r$同胚.

Let $q>0$ such that $[q[qn]]+1=[q^2n],n=1,2,\cdots$,solve $q$.

(1)计算$S^n(r)$上诱导度量$i^\ast g_0$在球极投影坐标下的表达式;

(2)计算$(S^n(r),i^\ast g_0)$的在一组正交标架下的联络形式;

(3)证明$(S^n(r),i^\ast g_0)$是常曲率黎曼流形.

(1) $f_{ij}=f_{ji}$,其中$i,j=1,2,\ldots,m$;

(2)$f_{ijk}-f_{ikj}=f_lR_{ijk}^l$,其中$i,j,k=1,2,\ldots,m$.

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