Articles of analysis

Show that the set is closed

Let $(E, d)$ be a metric space, $x$ element of $E$. Show that the set \begin{equation} A = \{y \in\ E : d(x, y) \geq 5 \} \end{equation} is closed. Generally, how would you go about this? I have an exam soon and I want to learn this.

$ S_n=\sum_{k=1}^n\frac{1}{k}$ then is $S_n$ bounded?

Let , $\displaystyle S_n=\sum_{k=1}^n\frac{1}{k}$. Which of the following is TRUE ? (A) $\displaystyle S_{2^n}\ge \frac{n}{2}$ for every $n\ge 1$. (B) $S_n$ is bounded sequence. (C) $\displaystyle|S_{2^n}-S_{2^{n-1}}|\to 0$ as $n\to \infty$. (D) $\displaystyle\frac{S_n}{n}\to 1$ as $n\to \infty$. As the series is divergent , so (B) is FALSE. If (C) is TRUE then $\{S_n\}$ is a Cauchy […]

A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

The title pretty much says it all. I am also looking for a concrete example if possible. I have looked at the proof, but I’m not exactly sure what it means because I am kind of confused on what the theorem actually says. Does it mean that if g(z) is a circle and f(z) is […]

A condition for a function to be constant

I need to proof this result: Let $\alpha >1$ and $c\in\mathbb{R}$. If $f:U\subset\mathbb{R}^m\rightarrow\mathbb{R}^n$, U open, satisfies $|f(x)-f(y)|\leq c|x-y|^\alpha$ for every $x$, $y$ $\in U$, then $f$ is constant in every component of $U$. I just didn’t have any idea on how to start it, I’m doing my first multivariable analysis course now!

find the limit: $\lim_{n\to\infty}\int_{0}^{\infty} \frac{\sqrt x}{1+ x^{2n}} dx$

Calculate the following limit $$\lim_{n\to\infty}\int_{0}^{\infty} \frac{\sqrt x}{1+ x^{2n}} dx$$ I tried to apply dominated convergence theorem but I could not find the dominating function. even I broke down the integration from $0$ to $1$ and $1$ to infinity. then found only integration from $0$ to $1$ is possible. Do you have any ideas?

Fourier series for $-x+\frac{1}{2}$

$[x]-x+\frac{1}{2}$ has the Fourier series $$\sum_{n=1}^{\infty} \frac{\sin{2n\pi x}}{n\pi}.$$ By evaluating the series directly, which requires some work, it can be shown that the series is convergent to $[x]-x+\frac{1}{2}$ if $x$ is not an integer. My question is, do we have any criteria by which we can easily see that $$[x]-x+\frac{1}{2}=\sum_{n=1}^{\infty} \frac{\sin{2n\pi x}}{n\pi}$$ when $x$ is […]

A Bound for the Error of the Numerical Approximation of a the Integral of a Continuous Function

How to numerically integrate a nasty function? Suppose $f$ is only continuos; which method can you employ to approximate $$\int_0^t f(s)ds$$ Since $f$ is continuos the integral exists, but all the numerical approximation methods I studied bound the error term with the hypothesis that $f$ is at least $C^2$ or something. I also know of […]

Proof that Laplacian is surjective $\mathcal{P}^n\to\mathcal{P}^{n-2}$

Let $\mathcal{P}^n$ denote the vector space of homogeneous polynomials on $\mathbb{R}^3$ of degree $n$. I need to prove that $\Delta|_{\mathcal{P}^n}:\mathcal{P}_n\to\mathcal{P}_{n-2}$, for $n\geq2$ is surjective, where $\Delta$ is the Laplace operator. The hint says that the proof should be done by inductive argument: in the inductive step I should conclude from the formula $$\Delta(x_1^{q_1}x_2^{q_2}x_3^{q_3})=q_1(q_1-1)x_1^{q_1-2}x_2^{q_2}x_3^{q_3}+q_2(q_2-1)x_1^{q_1}x_2^{q_2-2}x_3^{q_3}+q_3(q_3-1)x_1^{q_1}x_2^{q_2}x_3^{q_3-2}$$ that surjectivity […]

For any subset $S$ of $\mathbb{R}$, the distance function $d_s(x) = d(x,S)$ is continuous

Let $S \subset \mathbb R$ be any set, and define for any $x \in \mathbb R$ the distance between $x$ and the set $S$ by $d(x,S) = \inf\{|x-s| : s \in S\}$. Prove that the function $d_s: \mathbb R \to [0,+\infty)$ given by $d_s(x) = d(x,S)$, is Lipschitz continuous. Prove that if $S$ is compact […]

How to solve such a nonlinear ODE, the analytical solution of which is known!

I have the following ODE with initial/boundary value conditions: $$\left. \begin{aligned} \left(x^2-10 x-y^2\right)y\, y'(x)+(x-5) y^2 y'(x)^2-(x-5) y^2=0 ;\qquad (\text{ODE})\\ y(0)^2=25;\qquad y'(0)^2=\frac{3-\sqrt{5}}{2} \qquad\qquad\qquad (\text{IBCs}) \end{aligned} \right\} $$ How to solve such a nonlinear ODE? Additionally, how can I verify whether a special function is a potential solution or not, e.g., the one in implicit form as […]