臺灣大學數學系

八十八學年度第二學期碩博士班資格考試試題

 分析

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Part A: There are 7 problems in this section, do any 4 problems from them.

1. $x\in (0,1)$, Call $x$ a lucky number if there are infinite many 8's in the decimal expansion of $x$. Call $x$ a really lucky number if there are infinitely many 8's, infinitely many 88's, infinitely many 888's, infinitely many 8888's … in the decimal expansion. Show that the set of really lucky number is measurable and find its measure.
2. $f\in L^1(a,b)$, given $\varepsilon>0$, there exists $\delta>0$ such that $\int_E\mid f(x)\mid dx<\varepsilon$ whenever $m(E)<\delta$ for all measurable subset $E\subset(a,b)$. Where $m(E)$ is the Lebesgue measure of $E$.
   True or false? Justify your answer.
3. Let $f(x)=\sum_{n=0}^{\infty} {\cos nx\over 2^n}$
   (a) Does $f\in C^0(R)$ ?
   (b) Does $f\in C^1(R)$ ?
   (c) Does $f\in C^\infty(R)$ ?
   (d) Is $f$ analytic on $R$ ?
4. Let $f\in L^1(0,\pi)$, find the following limits:
   (a) $\lim_{n\rightarrow \infty}\int^{\pi}_0 f(x)\sin nxdx$
   (b) $\lim_{n\rightarrow \infty}\int^{\pi}_0 f(x) \mid\sin nx\mid dx$
5. (a) Is the closed unit ball in $l^2$ compact ?
   (b) Let $E=\{ (x_1,x_2,x_3,\cdots) \mid \ \mid x_i\mid \leq {1\over n}\}$. Is $E$ compact in $l^2$ ?
6. Does there exist a bounded linear operator $T: L^2(0,1)\rightarrow L^2(0,1)$. Whose range is the set of all polynomials ?
7. Show that $L^P(0,1)$ is complete for $1 \leq p<\infty$.

Part B: There are 4 problems in this section, do any 2 problems from them.

1. Evaluate the following integrals:
   (a) $\int_{\mid z \mid =r}{\mid dz \mid \over \mid z-a\mid ^2}$      (b) $\int_{\mid z \mid =r} {d\theta \over z-\alpha} \ , \ \mid \alpha \mid \ne r \quad (c) \int_{\mid z \mid \leq 1}{dxdy \over z-\alpha}$
2. If $f(z)$ is analytic in $\mid z \mid \leq 1$ and satisfies $\mid f(z) \mid=1$ on $\vert z\vert=1$. Show that $f(z)$ is a rational function.
3. Let Ω be the unit disk minus the real interval $(0,1)$. Find a harmonic function $u$ in Ω so that $u=1$ on $\mid z\mid=1$ and $u=0$ on $(0,1)$.
4. Let $C_1, C_2$, be two circles in the plane and suppose that $C_1$ lies inside $C_2$. Let $\Gamma_1$ be a circle in the annular region which is tangent to $C_1$ and $C_2$ somewhere and let $\Gamma_2$ be the circle tangent to $C_1, C_2$ and $\Gamma_1$; $\Gamma_3$ be circle tangent to $C_1, C_2$ and $\Gamma_2; \ \cdots$ . Suppose that the nth circle $\Gamma_n$ tangent to $\Gamma_1$ (as shown in the figure).
   Show that no matter how the first circle $\Gamma_1$ take place in the annular region, the last circle $\Gamma_n$ is always tangent to $\Gamma_1$ and $n$ is independent of $\Gamma_1$.

　

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