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

(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.

(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 $x, y \in \mathbb{R}^n$.