Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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$f:K \rightarrow K$ where $K$ is compact and $||f(x) - f(y)|| = ||x - y||$ implies $f$ is bijective

I found this exercise: $f:K \rightarrow K$ where $K$ is compact and $||f(x) - f(y)|| = ||x - y||$, for all $x, y$ in K, implies $f$ is bijective. Hint: show that $f$ is injective and continuous and ...
Arthur's user avatar
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2 answers
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$sin(1/x)$: expectation vs infimum

I want to show that $$ \underset{Q \in \mathcal{P}^0}{\inf} \ \{\mathbb{E}_{x \sim Q} \sin(1/x)\} > \inf_{x \in \mathbb{R}}\{ \sin(1/x)\} = -1$$ holds where $\mathcal{P}^0$ is the set of all ...
independentvariable's user avatar
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1 answer
26 views

Trivial cases of a Theorem of Criterion for Multiple Zeros

This is from Gallian's Contemporary Abstract Algebra: Theorem: A polynomial $f(x)$ over a field $F$ has a multiple zero in some extension E if and only if $f(x)$ and $f'(x)$ have a common factor of ...
baristocrona's user avatar
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1 answer
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Is $\int_0^{\pi}\log \sin \theta d\theta$ not well-defined?

Is $\int_0^{\pi}\log \sin \theta d\theta$ not well-defined? On the third edition of Ahlfors' Complex Analysis, page 160 it states: As a final example we compute the special integral \begin{equation*} \...
studyhard's user avatar
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Find the maximum value of $a$ under the condition $2e\ln x\leq ax+b\leq \frac{1}{2}x^2+e$,where $a,b\in\mathbb{R}$,$x>0$

Assume there exists $a,b\in \mathbb{R}$,such that $$2e\ln x\leq ax+b\leq \frac{1}{2}x^2+e$$ hold for $\forall x>0$.find the maximum value of $a$. I guess the critical situation is $y=ax+b$ it the ...
Ginkgo's user avatar
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Prove the following relationship of the number e and then show that it is irrational

I have been trying to show the following fact: $$e -(1+\frac{1}{1!}+ ... + \frac{1}{n!})<\frac{1}{nn!}$$ My steps: it seems reasonable to me to try to do this problem by induction, but when I tried ...
ruka's user avatar
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Continuity of this multivariable function, check

Can you please tell me if my reasoning is correct or if I missed some crucial step/point? I have to study if the following function is continuous $$f(x, y) = \begin{cases} \frac{x^2-y^2-1}{1-x} & ...
Martin and Friends's user avatar
1 vote
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Homogeneous function from monotonic functions

Consider a function of n variables $f(x_1,x_2...,x_n)$, stricly monotonic in each $x_i$ and $C^1$. Is it possible to construct a first-order homogeneous function from it of the form \begin{align} \...
cx1114's user avatar
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2 answers
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Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$

Let $f:[0,1]\to \mathbb R$ be a continous function on $[0,1]$ and differentiable on $(0,1)$, $f(0)=0, f(1)=1$. Prove that there exists $c\in (0,1)$ so that $f(c)+\frac{1}{e-1}=f'(c)$ where $e$ is the ...
user1259172's user avatar
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Find the potential $\Phi$ and the field $\mathbf{F} = \nabla \Phi$ for a two-dimensional dipole at the origin.

I'm struggling with one of the problems from mathematical analysis II course. The problem is next: Find the potential $\Phi$ and the field $\vec{\mathbf{F}} = \nabla \Phi$ for a two-dimensional ...
Nebula's user avatar
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Triple Integral Reiteration

I have the following triple integral: $$ I = \int_0^1 dz \int_z^1 dx \int_0^{x-z} f(x, y, z) \ dy $$ I want to reiterate the integral in such a way that the integrations are performed in the following ...
Nebula's user avatar
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prove that an interval of the form $(a,b]$ can be covered by open intervals of the form $(a_k,b_k)$

I need to prove that an interval of the form $(a,b]$ can be covered by open intervals of the form $(a_k,b_k)$. I tried to prove it using union of intervals of the form $(a_k,b_k + \frac{\epsilon}{2^k})...
a7777's user avatar
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Pointwise limit of functions on $[0,1]$

i was thinking on the problem bellow but I couldn't fully solve the problem, here is the statement: Let $f:[0,1] \rightarrow [0,1]$ be a continuous function such that $\forall x \in [0,1]$ there ...
Amir Mg's user avatar
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Asymptotics of Lambert W function

I want to show $$ e^{W(n+1)}-e^{W(n)} = o(1) $$ for $n\to\infty$. Since I'm not that used to the Little-o notation I'm wondering if my reasoning is correct. I tried to prove it in the following way ...
Ellenier's user avatar
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2 votes
2 answers
195 views

Proving a function isn't bounded

Let $f(x) = \frac{1}{x}$ be a function to and from the reals. Is $f$ bounded? I want to prove using the definition. Let $M > 0$ be given. Then there exists a $\delta > 0$ such that: If $0<|x-...
adisnjo's user avatar
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Concentration-compactness Lemma

Let $N\geq 3$ y $2^{*} := 2N/(N - 2)$. The space $\mathcal{D}^{1,2}(\mathbb{R}^N) := \left\lbrace u \in L^{2^{*}} (\mathbb{R}^N) : \nabla u\in L^2(\mathbb{R}^N)\right\rbrace $, with the inner product $...
M.J.519's user avatar
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Antiderivative of $g''(x)=f''(x)+\frac{1}{x^2}+e^{1-x}$

Let $f,g:(0,+\infty)$ be functions such that: (1) $g''(x)=f''(x)+\frac{1}{x^2}+e^{1-x}$, (2) $f(x) \geq g(x)-2x+2$, for all $x>0$. Show that $g'(1)=f'(1)+2$ and find the relation between $f$ and ...
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2 answers
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How to evaluate double integral: $\iint \frac{y}{x} \, dx \, dy$ if it is in the first quadrant and is bounded by: $y=0$, $y=x$, and $x^2 + 4y^2 = 4$

I want to evaluate this double integral: $$ \iint \frac{y}{x} \, dx \, dy \quad $$ which is bounded by functions: $$ y = 0 \quad $$ $$ y = x \quad $$ $$ x^2 + 4y^2 = 4 \quad $$ And is in the first ...
Nebula's user avatar
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1 answer
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Assumption of a partial derivative Lars Hörmander does in "The Analysis of Linear Partial Operators I"

In the proof Theorem 5.2.1 (equality (5.2.5)), Hörmander claims that for any smooth function $\psi \in C^{\infty} (\mathbb{R}^n)$ the following equality is true: $ \frac{\partial}{\partial \epsilon}(\...
Sebastian's user avatar
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Customizing the bump function

I have the standard bump function below. $$ \Psi(x) = e^{-\frac{1}{1 - \mathrm{min}(1, x^2)}} $$ How can I customize it to be like below: Translate and scale I can translate and scale by: $$ \Psi(x) ...
Megidd's user avatar
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1 answer
81 views

Differential of the function $\frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix} x \\ y \end{pmatrix} $

I have the following simple question find the differential: $D\phi(x;y) (h_1 ; h_2) $ of the function: $ \phi(x;y):= \frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix} x \\ y \end{pmatrix},$ defined on $\...
OffHakhol's user avatar
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2 votes
2 answers
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Convergence of $\sum_{n=1}^\infty \frac{\sin\frac{n^2+1}{n+1}}{\sqrt n}$ (again)

Already asked this question but I didn't get an answer and I still haven't made any progress. As I said before: I think the idea is to prove that sequence of partial sums of $\sum_{n=1}^\infty \sin\...
klop's user avatar
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For a vector-valued $f,$ is there a convex $\varphi$ such that $\nabla \varphi = f?$

Let $U \subset \mathbb{R}^d$ be a convex open set. Suppose a continuously differentiable vector-valued function $f : U \to \mathbb{R}^d$ satisfies that its derivative $D f : U \to \mathbb{R}^{d \times ...
Paruru's user avatar
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Relation between Bessel K functions

I am studying Bessel modified $K_\nu$ functions, and I do not find any clear reference for the following question: is there a relation between $K_\nu(x)$ and $K_\mu(x)$ in general? More specifically, ...
Wirdspan's user avatar
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Calculate the norm of $f$ in $\mathbb{C}[a,b]$

Let $X=\mathbb{C}[a,b]$ with the norm $\Vert x\Vert=\max_{a\le t\le b}|x(t)|$, If we have $t_1,\cdots, t_n\in [a,b]$ and $\lambda_1,\cdots, \lambda_n\in \mathbb{C}$. If and let $f(x)=\sum_{i=1}^n\...
S-fu's user avatar
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1 vote
1 answer
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Proof of the Chain Rule of multivariable functions in Munkres' Analysis on Manifolds

In Munkres' Analysis on Manifolds, page 57 Theorem 7.1(the Chain Rule) it states: ...For this purpose, let us introduce the function $F(\mathbf h)$ defined by setting $F(\mathbf 0)=\mathbf 0$ and \...
studyhard's user avatar
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Real Analysis, equation proof using binomial theorem

Can someone help me proving the following equation? enter image description here
Dary 's user avatar
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Where does 2.3.5 come from and how to prove it?

I was looking over this proof in Atkinson’s Numerical Analysis 2nd Edition and I am having a hard time seeing where 2.3.5 is coming from. What is Atkinson trying to show with 2.3.5 and the line below? ...
Dr. J's user avatar
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0 answers
31 views

Application of weak maximum principle.

Fix any open, bounded set $U \subset \mathbb{R}^n$ and suppose that $u \in C^2(U) \cap C(\bar{U})$ is a solution of $$-\Delta u = f \,\,\,\text{in}\,U$$ $$u=g \,\,\,\,\,\,\text{on}\,\,\,\partial U,$$ ...
Lilili123's user avatar
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Necessary condition for exponential stability of an LTV system

This is a homework problem from my adaptive control course: Consider the IVP $\dot x(t) = -u(t)^2x(t)$ with $x(0) = x_0$. Suppose the system is exponentially stable. Show that there exist some $\...
ArGenya's user avatar
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4 votes
0 answers
57 views

$L^p_{loc}(\Omega)$ is completely metrizable

Let $\Omega \subset \mathbb{R}^n$ be a (not necessarily bounded) domain and $1 \leq p \leq \infty$. Then define $L^p_{loc}(\Omega)$ to be the set of functions $f: \Omega \rightarrow \mathbb{R}$ such ...
CBBAM's user avatar
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0 votes
1 answer
53 views

Determining Conditions for the Exchange of Limit and Integral

I encountered this problem: Consider the following function: $$ f(x,\Delta k)=\lim_{\Delta k \to \infty} \frac{\sin ^{2}\frac{x\Delta k}{2}}{\Delta k\left ( \frac{x}{2} \right )^{2} }\equiv 2\pi\...
Lin Han's user avatar
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1 vote
0 answers
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Ask for solutions to two improper integrals whose integrands contain the integer powers of the sine function

Let $k\in\mathbb{N}$, $\alpha\in\mathbb{R}$, and $a,b\in\mathbb{R}$ such that $a,b>0$. Then \begin{equation}\label{alpha-n=1-k-odd}\tag{q1} \int_0^\infty\frac{\sin^{2k-1}t}{t}\sin(\alpha t)\...
qifeng618's user avatar
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0 votes
1 answer
102 views

How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously?

How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously? This equation is called “Gauss's Law” in physics. I seek for a rigorous mathematical proof for ...
studyhard's user avatar
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0 answers
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How do I apply the Riemann-Lebesgue lemma to prove the weak * convergence of this sequence of characteristic functions?

I am trying to prove the following Let $\Omega \subseteq \Bbb R^N$ be an open measurable set. For $\lambda \in (0, 1)$, consider a set $E \subseteq Q :=(0,1)^N$ with $|E| = \lambda|Q|$. For $n \in \...
darkside's user avatar
  • 585
3 votes
1 answer
92 views

Can Darboux Theorem be stronger by making the derivative at c continuous?

Darboux Theorem: If $f$ is differentiable on $[a,b]$ and $\eta$ between $f'(a)$ and $f'(b)$, then exists $c\in[a,b]$ such that $f'(c)=\eta$. The common counterexample about $f'$ at $c$ not necessarily ...
ZhenRanZR's user avatar
1 vote
1 answer
92 views

Validity of the proof $\mathbb R$ is uncountable

My attempt: take the interval $H=(0,1)$. Assume $(0,1)$ is countable, then there exist a bijection from $\mathbb N$ to $H$ where. Define elements of $H$ as a monotonically increasing sequence $\{a_i\}...
user1259172's user avatar
-3 votes
0 answers
21 views

Why circular domain could not be reinhardt domain,what's the difference between them? [closed]

complex analysis enter image description here
sen wang's user avatar
1 vote
2 answers
59 views

Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq 1,y\leq1,-1\leq z\leq 1\}$ a star domain?

Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq1,y\leq1,-1\leq z\leq1\}$ a star domain. My approach: Let be $N:=\{(x,y,z)\in\mathbb{R}^3\mid x\leq1,y\leq 1,-1\leq z \leq1\}$ and $M:=\...
Philipp's user avatar
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-1 votes
0 answers
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is a ball in n-dimentions a set? [closed]

Can you say that a ball in n-dimention space is a set of all dots that are at the distance R or closer to the center of a ball?
Nila Alesich's user avatar
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0 answers
46 views

Given $ \frac{1}{a_{n+2}} - \frac{1}{a_{n+1}} = e^{-(a_{n+1}-b)} \left( \frac{1}{a_{n+1}} - \frac{1}{a_n} \right)$ [closed]

Given the recursive relation $$ \frac{1}{a_{n+2}} - \frac{1}{a_{n+1}} = e^{-(a_{n+1}-b)} \left( \frac{1}{a_{n+1}} - \frac{1}{a_n} \right), \quad a_0, a_1, b \in [0, \infty). $$ It has been proved ...
murph's user avatar
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-1 votes
0 answers
31 views

Is the real coordinate n-space a field? [closed]

Im studying real analysis on my freshman year. And we've been introduced to two concepts: a field and a metric space. And its a bit confusing - is a metric space a field? is Rn a field and a metric ...
Nila Alesich's user avatar
1 vote
1 answer
52 views

Are convergent sequences closed under uniform convergence?

Setting: Let $Y$ be a metric space and let $a_{n,k}\in Y$ for all $n\in\mathbb{N}$ and $k\in\mathbb{N}$. Suppose $a_{n,k}\to a_{\bullet, k}$ uniforly as $n\to\infty$. Suppose the sequences $\{a_{n,k}\...
John Frank's user avatar
-2 votes
0 answers
39 views

A finite situation that arises from the Baire Category Theorem [closed]

Suppose $[0,+\infty) = F_1\cup F_2$ where $F_1$ and $F_2$ are closed sets (with the usual (subspace) topology of $\mathbb{R}$). Is it true that $0$ belongs to the interior of $F_1$ or $F_2$ ? A ...
Alex's user avatar
  • 1
2 votes
1 answer
58 views

L'Hopital's rule with dual numbers

Background: For the dual numbers, we extend the reals with an additional unit vector $\epsilon$ subject to the constraint that $\epsilon^2 = 0$. We can write dual numbers as $x_0 + x_1 \epsilon$ for $...
kc9jud's user avatar
  • 236
5 votes
0 answers
73 views

Solving the limit of a pretty ugly looking integral [duplicate]

Calculate the following limit $$\lim_{n \to \infty} \int_0^\pi \frac{2(2\sin{x}-\sin{((2n+1)x)}+\sin{((2n-1)x)}}{5-4\cos{(2nx)}-\cos{(4nx)}}dx$$ After some trigonometry manipulation (writing the ...
Shthephathord23's user avatar
3 votes
2 answers
93 views

Proving $e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu}+\left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2}$

I'm trying to show that $$e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu} + \left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2},$$ for an absolute constant $C &...
stoic-santiago's user avatar
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0 answers
55 views

$z \mapsto z^m$ is orientation preserving and regular

I'm following GP's book of Differential Topology and I stuck in the proof of Fundamental Theorem of Algebra using Intersection Theory. We have the afirmation of $f: \mathbb{S^1} \to \mathbb{S}^1, s.t.,...
junior_eyes1313's user avatar
1 vote
1 answer
48 views

Confusion about Limits (Rationals)

$ f(x)= \begin{cases} x^2&\text{if $x$ is irrational}\,\\ 2x+1&\text{if $x$ is rational}\\ \end{cases} $ I want to calculate the limit of $f(x)$ as $x$ tends to $0$. Is it enough to just say ...
adisnjo's user avatar
  • 59
0 votes
1 answer
163 views

Prove $\lim_{x\rightarrow +\infty}f(x)=0$ under the conditions $|f'(x)|\leq 1/x$ and $\lim_{R\rightarrow +\infty}\frac{1}{R}\int_0^R|f(x)|dx=0$ [closed]

I encountered a problem while working on a mathematical analysis exercise. I wish to share this problem because of its interest, while it is a simple statement I feel it is not straightforward to ...
Liping Li's user avatar

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