臺灣大學數學系

八十七學年度第一學期碩博士班資格考試試題

 分析

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(前六題任選四題, 後二題任選一題)

一.

a.

Let $\{ f_n \}$ be a sequence of real-ralued Lebesgue integrable functions on an interval $[a, b]$. State and prove sufficient conditions for $\int_a^b \lim f_n(x) dx$ and $\lim \int_a^b f_n(x) dx$ to exist and be equal.

b.

Let $\{ f_n \}$ be a sequence of real-ralued functions and continuously differentiable on $[a, b]$. State and prove sufficient conditions for $\lim f_n^{\prime} (x)$ and $( \lim f_n )^\prime (x)$ to exist and be equal for $x \in [a, b]$.

二.

If $f(x)$, $x \in R$, is continuous at almost every point of an interval $[a, b]$, show that $f$ is measurable on $[a, b]$.

三.

a.

If $f$ is measurable on $E \subset \mathbb{R}^n$, show that

$$\vert \{ x \in E : f(x) > \alpha \} \vert \leq {1 \over {{\al... ...lpha \} } f^p (x) \ dx \,,\,\, \alpha > 0,\, 0 <p < \infty \,.$$

b.

Let $f$ and $\{ f_n \}$ be measurable functions on $E$. If $p > 0$, and $\int_E {\vert f-f_n \vert}^p\ dx \rightarrow 0$ as $n \rightarrow \infty$, show that $f_n \buildrel m \over \longrightarrow f$ on $E$ (converge in measure on $E$).

四.

Let φ be a convex function on $(- \infty, \infty)$ and $f$ an integrable function on $[0, 1]$. Show that

$$\int_0^1 \varphi (f(t))\ dt \geq \varphi \big( \int_0^1 f(t) \ dt \big)\, .$$

五.

Let $f$ be a Lebesgue integrable function on $[0, 1]$ and assume that $\int_0^1 f(x) \varphi (x)\ dx =0$ for all continuous functions $\varphi : [0, 1] \rightarrow R$. Show that $f=0$ almost everywhere.

六.

Let φ be a positive continuous function on $R$ satisfying $\varphi (x)=0$ for $\vert x \vert >1$ and $\int_{-\infty}^{\infty} \varphi(x)\ dx =1$. Defint $\varphi_n (x) =n \varphi (nx)$, $n=1, 2, \cdots .$ Show that if $f$ is a continuous function on $R$, then $\int_{-\infty}^{\infty} \varphi_n (x-y)f(y) \ dy$ converges to $f(x)$ as $n \rightarrow \infty$ for all $x \in R$.

七.

Let $f$ be a function which is analytic in an open set containing the closed disc $D=\{ z : \vert z \vert \leq 1 \}$. Show that the points where $Re(f)$ (the real part of $f$) takes its maximum in $D$ are on the boundary of $D$, and show the same for minimum points.

八.

Find the Laurent expansion for the function $f(z)={{(z+2)e^z}\over z^3}$ in powers of $z$. In what region of the complex plane does this series converge. Calculate the integral $\oint_{\{ \vert z \vert =1 \}} f(z)\ dz$, where the circle $\vert z \vert =1$ is travelled in the counterclockwise direction.

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