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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1
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1answer
21 views

Does the airy group “commute” with time-compactly supported functions?

I am currently reading the book "Introduction to nonlinear dispersive PDEs" by Felipe Linares and Gustavo Ponce and found an equality that really kept my attention. At the top of page 173 (...
4
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2answers
237 views

Showing that Lebesgue Dominated convergence theorem is false in case of Riemann integration.

I was reading Tom Apostol book called "Mathematical Analysis" and I read this statement: the Lebesgue Dominated convergence theorem is false in case of Riemann integration. Here is the ...
1
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1answer
52 views

What is area using a double integrals between the curves?

What is area enclosed between curve/coordinate axes? $$(\frac{x}{a})^3+(\frac{y}{b})^3=\;(\frac{x}{h})^2+(\frac{y}{k})^2\;,(x=0,y=0); $$ $ (a,b,h,k)$ are constants. And $ x,y \geq 0$
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0answers
34 views

Integrability of $\frac{|xy|^a}{nz^3}$ over $\sqrt{x^2+y^2}<z<\sqrt{n^2-x^2-y^2}$

Let $a>0,$ $$f_a(x,y,z)= \frac{|xy|^a}{nz^3}$$ on $$A_n=\{(x,y,z):\sqrt{x^2+y^2}<z<\sqrt{n^2-x^2-y^2}\}$$ I want to find all $a>0$ such that $f_a \in L^1(A_n)$ for $n \geq1.$ Looking at ...
24
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3answers
369 views

Prove or disprove the existence of $A$

I want to know if there exists a set $A \subseteq \mathbb{N}$ such that $$ \lim_{x\to\infty} x^2 e^{-x} \sum_{n\in A} \dfrac{x^n}{n!}=1 $$ More generally, the question will be the existence of a set $...
2
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0answers
53 views

On the convergence of the function series $\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$

Let $f$ be a smooth real function defined around origin. Consider the series $\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$. If we differentiate term by term the series, we get $\frac{...
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1answer
37 views

Alternative reading materials for Baby rudin after chap.8

Next semester I will study chapter 9, 10, and 11 of Baby Rudin. However, from Chapter 9, I was told that understanding the part of analysis (like functions of several variables, integration of ...
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1answer
31 views

Finding the local extrema of $y = \frac{\ln x} {\sqrt{x}}$

I'm given a function: $$y = \frac{\ln x} {\sqrt{x}}$$ I've found the first derivative, which is $$y' = \frac{2-\ln(x)}{2x\sqrt(x)}$$ and got that the domain is $x > 0$, but I just don't know how to ...
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0answers
52 views

Perturbing the Initial Condition of an ODE within a compact

Let $\gamma: (a,b) \rightarrow \mathbb{R}^d$ be the maximal solution of the differential equation $x' = F(t,x)$ with initial condition $\gamma(t_0)=x_0$. Show that given any compact $K \subset I$ and ...
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1answer
30 views

The interval $(a,b) \subseteq \mathbb{R}^{2}$ is bounded - metric spaces

I am trying to show that the interval $(a,b) \subseteq \mathbb{R}^{2}$ is a bounded set. By $(a,b) \subseteq \mathbb{R}^{2}$ I am meaning $(a,b) \times \{0\} = \{(x,y) \in \mathbb{R}^{2}: a<x<b, ...
2
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0answers
29 views

find isolated local extrema of following function f

I have problems with the following: we have function $h\colon \mathbb{R}\to\mathbb{R}$, $h(x)= \begin{cases} -1 & x<-1\\ \sin(\frac{\pi}{2} x) & -1\le x\le 1\\ 1 & x>1 \end{cases}$ ...
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5answers
75 views

How can you prove the derivate of $e^x$

My question is related to the prove that $\frac{dy}{dy}e^x=e^x$ I now a prove, but I think is can be applied to any $\alpha^x$, therefore I want to know why it does not work like that. The proof $$\...
2
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3answers
235 views

Why does L'Hospital's rule require the limit to exist? About the proof.

Proof of L'Hospital's rule (only special case this time): Let's assume, that $$ f(a)=g(a)=0 .$$ Using the MVT, we get $$ \frac{f(x)}{g(x)}=\frac{f(x)-f(a)}{g(x)-g(a)}=\frac{f'(\zeta)}{g'(\zeta)}, \ \...
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1answer
19 views

Showing that $\mathbb{C}$ is closed, open, perfect but not bounded - metric spaces

Consider the following subset of $\mathbb{R}^{2}.$ The set of all complex numbers (i.e. $\mathbb{R}^{2}$). I'm trying to show that this set is Closed, Open, Perfect but not Bounded. Closed: let us ...
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1answer
12 views

extrema under constraints - lagrangian multipliers

Find all global extrema of $f(x,y,z)=x^3+y^3+z^3$ under the constraints a) $x^2+y^2+z^2=1$ b) $2x^2+y^2=1$ Regarding a), I've tried to use the lagrangian multipliers, that is $(3x^2-\lambda 2x,3y^2-...
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2answers
38 views

Proving $\mathbb{Z} \subseteq \mathbb{R}^{2}$ is closed, not open and not bounded metric spaces

I'm working my way through the examples in Rudin on open sets, closed sets and other related material to get use to working with the definitions. I'm trying to show that $\mathbb{Z} \subseteq \mathbb{...
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2answers
35 views

Understanding the proof of Proposition 10 in Ch.2 in Real analysis by Royden and Fitzpatrick “Fourth Edition ”

Here is the proposition and its proof: My question is: I do not understand how the last equality came from the one just before it, could anyone explains this for me, please?
2
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1answer
53 views

Show that function distances are preserved

If for all $x,y \in \mathbb{R}^n$ that satisfies $|x - y| = t$ also satisfies $|f(x) - f(y)|=t$ (for some constant $t \in \mathbb{R}^{+}$), show that $|f(x) - f(y)| = |x-y|$ for all values of $x,y \in ...
0
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1answer
21 views

Identity for the space-time fourier transform of the “forced term” in the airy equation

Let's consider a function $f(t,x)$ smooth and sufficiently fast decaying in $\mathbb{R}^2$, where $t,x\in\mathbb{R}$. Consider also the time-dependent operator $V(t)$ given by $$ V(t)g:=\int_{-\infty}^...
1
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1answer
21 views

$L_p(\mu,X)$ is isometrically isomorphic to $\ell_p(X)$

In the book Banach Spaces of Vector-Valued Functions the authors present a demonstration for Proposition 1.6.4, pages 32-33: Let $1\le p \le + \infty$, if $(\Omega,\Sigma,\mu)$ is a $\sigma$-finite ...
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1answer
50 views

What can be said about the convergence of the following integral? [closed]

What can be said about the convergence of the integral $\int_{\frac{\pi}{2}}^{\infty} \frac{1}{1+\sin x}dx$?
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1answer
37 views

Show that $f(x):=\sum\limits_{n=0}^{\infty}\frac{1}{n}h(2^{n}x),$ where $h$ is a piecewise function, converges uniformly on $[0,1]$

For $x\in\mathbb{R}$, consider a piecewise function defined by $$h(x):=\left\{ \begin{array}{ll} x,\ \ \ 0\leq x\leq 1\\ 2-x,\ \ 1\leq x\leq 2\\ 0,\ \ \text{otherwise}. \end{array} \right.$$ Now, ...
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0answers
20 views

periodic orbit of a system

Given $f(x,\lambda)$, $C^1$ in $\mathbb{R^2}\times \mathbb{R}$ that $x'=f(x,0)$ has a periodic solution $p(t)$. Suppose that $\omega$ is the period and the unique solutions of the linear system $$y'=...
0
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1answer
45 views

Integrability of $f(x,y)=\frac{(x+y)\log(y/x)}{(xy)^{a}(1+x^2+y^2)}$ on $O=\{(x,y):x>0,y>0\}.$

Let $$f_a(x,y)=\frac{(x+y)\log(y/x)}{(xy)^{a}(1+x^2+y^2)}=\frac{1}{x^a}\frac{(x+y)\log(y/x)}{y^a(1+x^2+y^2)} $$ defined on $$O=\{(x,y):x>0,y>0\}.$$ I need to find all $a\in \mathbb{R}$ such that ...
0
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2answers
60 views

Doubt about substitution in $\int_0^{2\pi} \frac{dx}{2+\cos x}$

While evaluating $$\int_0^{2\pi} \frac{dx}{2+\cos x}$$ I thought about letting $t=\tan \frac{x}{2}$, but I get the obviously wrong result $$\int_0^{2\pi} \frac{dx}{2+\cos x}=\int_0^{\tan\pi} g(t)dt=\...
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2answers
79 views

Prove that $\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$ [closed]

Let $f:[0,\infty)$ be Lebesgue-integrable, then prove that $$\lim_{\lambda\rightarrow\infty} \frac{1}{\lambda}\int_0^\lambda\int_0^xf(y)\,dy\,dx = \int_0^\infty f(x)\,dx$$ This is also known as Cesàro ...
0
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1answer
30 views

Show that the function $f(x)g(x)$ is integrable(2).

As a completion of this question here Show that the function $f(x)g(x)$ is integrable. I do not know how to answer $(b)$ and $(c)$ below. Let $A:=[a,b].$ Suppose that the function $f: A \rightarrow \...
0
votes
1answer
19 views

Dense set containing an orthonormal basis

It is known that if $E$ is a dense subspace of a separable Hilbert space $H$, then there exists an orthonormal basis of $H$ which is contained in $E$. I don't believe this is true for just a dense ...
1
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1answer
19 views

Question regarding to a step of a proof of implying countable cases from finite cases

I have a question regarding to a specific step of the following proof. I am confused to how did $\sum$ for all n implies $\sum $ for countable cases? My initial thought is math induction. However, ...
1
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0answers
37 views

An open in Banach space if connected if and only if is path-connected $C^{\infty}$

To prove the that An open in Banach space if connected if and only if is path-connected $C^{\infty}$ I'd like to prove the following : $\textbf{Lemma :}$ Let $U$ be open in a Banach space and $D \...
0
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6answers
50 views

Prove the left hand side of the following inequality

$\frac{2ab}{a+b} \le \sqrt{a \cdot b} \le \frac{a+b}{2}$ where $a,b > 0$ The right hand side $\sqrt{a \cdot b} \le \frac{a+b}{2}$ is the AM-GM inequality, it's clear how to solve it. Does the left ...
1
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1answer
51 views

Integrability of $f(x,y)=(1-x)^{a}$ on $D=\{(x,y):0<x^2+y^2<1\}$

As it is said in the title, I need to check the integrability of $$f(x,y)=(1-x)^{a}$$ for $a\in \mathbb{R}$ on the open disk $D= \{0<x^2+y^2<1\}.$ My attempt Let $a \geq 0,$ then since $D \...
0
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3answers
28 views

A sequence of $x_n \in (0,1)$ such that $x_n \to 1$ with explicit $|x_n-x_k|$.

I am looking for a sequence of $x_n \in (0,1)$ such that 1.$x_n \to 1$ we have an explicit form for $|x_n-x_k|$ for every $n$ and $k
0
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1answer
10 views

Stock Technical Analysis: Keltner Channel Calculation

I'm a software engineer and I'm not so good at maths, I am writing some software which performs technical analysis on stocks but it appears my maths is slightly off and I have spent hours and hours ...
1
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1answer
48 views

Show that the function $f(x)g(x)$ is integrable.

Let $A:=[a,b].$ Suppose that the function $f: A \rightarrow \mathbb{R}$ is continuous, $g: A \rightarrow \mathbb{R}$ is integrable and $g(x) \geq 0$ for almost all $x \in A.$ $(a)$ Show that the ...
0
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1answer
16 views

A sequence on $\{x_n\} \subset (0,1)$ such that $\lim_{n \to \infty} x_n=1$ with nice minimum distance properties.

I am looking for a nice example of sequence $\{x_n\} \subset (0,1)$ such that $\lim_{n \to \infty} x_n=1$. $|x_n-x_k|= f(|n-k|)$ for some explicit function $f$.
0
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1answer
30 views

Prove that the 2 sequences are nested intervals and give the element of the nested interval

$$ a_{n+1} = \frac {2a_{n}b_{n}}{a_{n}+b_{n}}, b_{n+1}= \frac{a_{n}+b_{n}}{2}$$ where $0<a_{1}<b_{1}$ and $n \in \mathbb{N}$ and proven is that the sequence $[a_{1} b_{1}], [a_{2}, b_{2}]... $...
0
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2answers
20 views

Proving equation has solution for every $c ≥ 0$

Task: Proof that the equation $x^5 − x = c$ has a solution for every $c \ge 0$ in the interval $[0, \infty)$. No idea where to start, anyone have any suggestions? Kind regards Anthony
0
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1answer
10 views

Tangent plane equation in the point of intersection with $\space y$-axis

I've encountered one tiny problem - I have to find tangent plane equation of the function $\space f(x,y)=y^3 - \sqrt{1-x^2y^2} \space$ in the point where it intersects with $\space y$-axis. So $\...
0
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2answers
37 views

Show that $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$.

I am trying to prove that this infinite series $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$. I can definitely show that this series converges ...
0
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0answers
21 views

Distance between subparts of a metric space and their intersection problem

Let $(G, \delta)$ be a metric space. Let $A, B$ two subparts of $G$ We pose $d(A, B)=\inf _{a \in A, b \in B} \delta(a, b)$ Show that if $A \cap B \neq \emptyset$ then $d(A, B)=0$ : My answer is ...
0
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0answers
30 views

solution space of complex ODE

Let $A\in\mathbb{C}^{n\times n}$ be a matrix, $\omega\in\mathbb{C}$ not be an eigenvalue and $p:\mathbb{R}\to\mathbb{C}^n$ be a polynomial function with coefficients in $\mathbb{C}^n$. Let $\psi:\...
0
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1answer
49 views

Inequality proof including $n!$

$n^{n}e^{-n+1} \le n! \le n^{n}e^{-n+1} n$, $n \in \mathbb{N}$ I'm struggling solving the inequality above, I have tried AM-GM, Bernoulli but I guess now that the proof is based maybe on induction.The ...
1
vote
1answer
61 views

How Lebesgue integration solved the problem of a function being integrable but its limit is not integrable?

My professor gave us the following form of Dirichlet function as an example of the problems we faced in Riemann integration: $\{r_{n}\}$ enumeration $\mathbb{Q} \cap [0,1]$ $$ f_{n}(x) = \begin{cases} ...
4
votes
2answers
58 views

Prove that the integral $\int_a^b \frac{\sin(x)}{x}dx$ is bounded uniformly.

How do I show that exists a constant $M>0$ such that, for all $0\leq a \leq b < \infty$, $$\left|\int_a^b \frac{\sin(x)}{x}dx\right| \leq M.$$ I just read on Richard Bass book's that is enough ...
1
vote
1answer
31 views

Why is $(1 - \frac{1}{n^{1-\epsilon}})^{n} < e^{-n^{\epsilon}}$ for $0 < \epsilon < 1$?

This argument appears in one proof in my lecture and I don't know why this holds. Maybe someone knows a theorem that implies this inequality? Thanks for help.
0
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1answer
34 views

Lower bound for Jacobi determinant

Consider a map $s\in C^1(Q,\Omega)$ where $Q,\Omega\subset\mathbb{R}^2$ are bounded and have Lipschitz boundary. Moreover $s$ is invertible and $||s'||_{L^{\infty}}<c$ ,$||(s')^{-1}||_{L^{\infty}}&...
3
votes
1answer
64 views

The least distance of $f\in\ell_\infty(K,\mathbb C)$ from $C(K,\mathbb C)$

Suppose that $K$ is a compact Hausdorff space. Consider a bounded function $f:K\to\mathbb R$ not necessarily continuous, that is, $f\in\ell_\infty(K,\mathbb R)$. It's a well-known fact that the least ...
1
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2answers
39 views

Show that $ x\cdot\cos(x)+\sin(x)/2=\sum_{n=2}^\infty (-1)^n\cdot\frac{2n}{n^2-1}\cdot\sin(nx)$ when $x\in [-\pi,\pi]$

To show this, I have used definitions for $\cos(x)$ and $\sin(x)$: $$x\cdot \cos(x)+1/2\sin(x)=x\cdot \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\cdot x^{2n}+1/2\cdot \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!...
0
votes
2answers
30 views

Verification of a logarithmic inequality

Verify the inequality $ \frac{(\log (x) + \log (y))}{2} \le \log\frac{(x+y)}{2}$, where $x,y>0$ I'm still struggling how to solve the inequality, I have tried AM-GM and Bernoulli, without any ...

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