102年 - 102 國立高雄大學_博士班招生考試試題_應用數學系(一般生、在職生)：高等微積分#147993

科目：研究所、轉學考(插大)-高等微積分 | 年份：102年 | 選擇題數：0 | 申論題數：10

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所屬科目：研究所、轉學考(插大)-高等微積分

選擇題 (0)

申論題 (10)

(1) (10 %) Show that, for any $m, n \in \mathbb{N} \cup \{0\}$,

$$\frac{1}{2\pi} \int{-\infty}^{\infty} \int{-\infty}^{\infty} \int{-\infty}^{\infty} (x + \mathrm{i}y)^m (x + \mathrm{i}z)^n e^{-\frac{1}{2}(x^2+y^2+z^2)} \, dx \, dy \, dz = \sqrt{2\pi} \delta{m,n} \cdot n!,$$

where $\mathrm{i} = \sqrt{-1}$ and $\delta_{m,n} = 0$ as $m \neq n$; 1 as $m = n$.
(2) \((10\%)\) Suppose that the coefficients of the power series \(\sum a_{n}z^{n}\) are in integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most 1.
(3) \((10\%)\) \(f\) is continuous on \([0,1]\) and if \(\int_0^1 f(x)x^n dx = 0\) for any \(n = 0,1,2,\ldots\), prove that \(f(x) = 0\) on \([0,1]\).
(4) \((18\%)\) Calculate the following integrals and limits:

(a) $\int0^1 \frac{\ln(1+x)}{\ln(1+x)+\ln(2-x)} \, dx$ (b) $\lim{n\to\infty} \sqrt[n]{2^n + 3^n + 5^n + 7^n}$ (c) $\lim_{n\to\infty} \frac{\sqrt[n]{n!}}{n}$.
(5) (10 %) If $E$ is a nonempty subset of a metric space $X$, define the distance from $x \in X$ to $E$ by $\rhoE(x) = \inf{z \in E} d(x, z)$. Prove that $\rho_E$ is a uniformly continuous function on $X$, by showing that

$$|\rhoE(x) - \rhoE(y)| \leq d(x, y)$$

for all $x, y \in X$.
(6) (10 %) Suppose that $f$ is Riemann integrable on $[0, A]$ for all $A < +\infty$, and $f(x) \to 1$ as $x \to +\infty$. Prove that

$$\lim{t\to 0} t \int0^{\infty} e^{-tx} f(x) \, dx = 1 \quad (t > 0).$$
(7) \((10\%)\) Let \(B = \{f\in C([0,1],\mathbb{R}):f\) is \(C^1\) on \([0,1]\), \(f(0) = 0\), and \(|f'(x)|\leq 1\}\). Show that the closure of \(B\) in \(C([0,1])\) with the supremum norm is compact.
(8) \((10\%)\) Let \(f{n}:[0,1]\to \mathbb{R}\) be defined by \(f{n}(x) = \frac{x^{2}}{(1 + x^{2})^{n}}\)

(a) Prove that \(\sum{n=1}^{\infty} fn(x)\) is convergent for \(x \in [0,1]\).
(b) Is it uniformly convergent? Explain your answer.
(9) (12 %) $A$ is an $n \times n$ symmetric real matrix and $f(x) = \langle Ax, x \rangle$, $x \in \mathbb{R}^n$, where $\langle x, y \rangle = \sum{i=1}^n xi yi$ for $x = (x1, \ldots, xn)$ and $y = (y1, \ldots, y_n)$ in $\mathbb{R}^n$. Show that $f$ is differentiable and find $Df(x)y$ for $x, y \in \mathbb{R}^n$.