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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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1answer
7 views

Convergence of the Cauchy product of two series of real numbers

During these days, I reading a paper about analysis. I am confused about following question of a Cauchy-product of two convergent series. Let $\sum_{n=0}^{\infty}a_n$ and $\sum_{n=0}^{\infty}b_n$ be ...
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0answers
16 views

Is my calculation correct?

For $\Omega=C_{\beta} \subset \mathbb{R^2}$ and $\beta \in(0,2\pi)$ define $$C_{\beta}=\left\{(r\cos(\phi),r\sin(\phi))\in\mathbb{R^2}|0<r<1,|\phi|<\frac{1}{2}\beta\right\}$$ and $\hspace{...
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0answers
19 views

Approximate countable union of open intervals by a finite covering of open intervals.

Given an open interval $I=(a, b)\subset \mathbb R$ for some $a, b\in \mathbb R$ and a countable union of its open subintervals $U=\bigcup\limits_{i=1}^\infty J_i$ such that $\sum\limits_{i=1}^\infty L(...
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0answers
7 views

linearized pde problem [on hold]

How I can linearization a equation with laplace operator
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0answers
25 views

What is the asymptotic behavior of a solution of this ODE?

Consider this ODE: $$f_{i}’(r) = f_{i}^2 (r) + O\left(\frac{1}{r^4} \right)$$ As $r \to \infty$. $i=1,2$ and the initial conditions are $f_{i} (1) = C_{i} < 0$ This is equivalent to the ODE: $$...
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0answers
32 views

Is this only definition?

If $u$ is twice differentiable and $\Omega$ a domain and $\Delta$ the Laplace operator then $$ -\Delta u \in C(\Omega) \Rightarrow u \in C^2(\Omega) $$ I think it is only definition . Thanks ...
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0answers
17 views

Characterization of summable families

Could somenone help me with to prove the following result? If $C=\{\lambda\in{\Lambda}:x_\lambda\neq{0}\}$ is countable and for every bijective mapping $\tau:\mathbb{N}\longrightarrow{C}$ $\sum_{n\...
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0answers
23 views

Does this proof of continuity at irrationals of the Thomae function make any sense?

Thomae's function $ f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = 0$ if $x$ is irrational and $f(x) = \frac{1}{q}$ if $x = \frac{p}{q}$. The function $f$ is continuous at all the irrational ...
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0answers
17 views

How to find: $f^{\alpha}_n(x)=1+\sum_{k=1}^{\infty} \frac{\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$

I am looking for a closed-form solution to $$f^{\alpha}_n(x)=a_0+\sum_{k=1}^{\infty} \frac{a_k\Gamma(n+\alpha)}{\Gamma(k+\alpha)\Gamma(n-k+\alpha)} x^k$$ where we can take $a_k = 1, \forall k\in [0,\...
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1answer
25 views

Limiting behavior of a meromorphic function

Refer the following answer. Can someone explains how can this follows from Morera's theorem. Thanks in advance.
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0answers
49 views

Zeros of $ f''$

Let $ f : \mathbb{R} \to \mathbb{R} $ be a $C^2$ function such that $$ \lim_{x \to \pm \infty}{f(x)} = 0 $$ Prove that $f''$ has at least two zeros. Assume $f$ is not a constant. Than $f$ must ...
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0answers
19 views

Prove $φ_n =2^{-1+2n}(-1+2^n)\to f$ pointwise

(Particular case of the general theorem and proof) Let $f:X\to[0,\infty]$ be measurable function defined on the interval $(k2^{-n},(k+1)2^{-n}]$ and $φ_n:X\to [0,\infty)$ monotone increasing ...
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1answer
31 views

The breakdown of the solution to the ODE $y' = \sqrt{4-2y}$.

Suppose $y=y(x)$ be a real valued function such that $$ y(x) = \int_0^x \sqrt{4-2y(t)} dt. $$ By differentiating both side with respect to $x$, we get the ODE $$ y' = \sqrt{4-2y}, \quad y(0)=0. $$ ...
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1answer
42 views

How to find $f(x)$ if $\int_0^{x^2} (1+t)f'(t)dt=x^4$ and $\int_0^1 f(t)dt=3 $

Knowing that $f:[0,+\infty)\rightarrow \mathbb{R}$ is continuous and derivable, and that: $\int_0^{x^2} (1+t)f'(t)dt=x^4$; $\int_0^1 f(t)dt=3 $ Determine $f(x)$. (Note: This is supposed ...
2
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1answer
42 views

Suppose $f : [0, 1] \to [0, 1]$ is differentiable such that $|f'(x)|\ne1$ for all $x\in[0, 1]$. Prove $\exists !a,b\in[0,1]:f(a)=a,f(b)=1-b$

Suppose $f : [0, 1] \to [0, 1]$ is differentiable such that $|f'(x)|\ne1$ for all $x\in[0, 1]$. Prove there exist unique $a,b\in[0,1]$ such that $f(a)=a$ and $f(b)=1-b$. Not sure how I'm ...
4
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1answer
82 views

Find function $f(x)$ satisfying $\int_{0}^{\infty} \frac{f(x)}{1+e^{nx}}dx=0$

I am looking for a non-trivial function $f(x)\in L_2(0,\infty)$ independent of the parameter $n$ (a natural number) satisfying the following integral equation: $$\displaystyle\int_{0}^{\infty} \frac{f(...
2
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0answers
11 views

Bounded linear operator $A$ s.t. $Ax=y$ has a least square solution for each $y$ iff the range of $A$ is closed

Let $A: H_1\to H_2$ be a bounded linear operator, where $H_1,H_2$ are Hilbert spaces. Prove that $Ax=y$ has a least squares solution for each $y\in H_2$ if and only if the range of $A$ is closed. I ...
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0answers
27 views

A question regarding inequality between weighted averages

Suppose you have three probability density functions $p_{1}(x)$, $p_{2}(x)$, and $p_{3}(x)$. Suppose further that the expectation values of $p_{1}(x)$ and $p_{2}(x)$ are unequal $\int_{0}^{1}p_{1}(x) ...
1
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1answer
60 views

$y_0 \ge 2$, $y_n = y_{n-1}^2 -2$ $\Rightarrow$ $\frac{1}{y_0}+\frac{1}{y_0y_1}+\cdots = \frac{y_0 - \sqrt{y_0^2 - 4}}{2}$

$y_0 \ge 2$ and $y_n = y_{n-1}^2 - 2$. Let $S_n = \frac{1}{y_0} + \frac{1}{y_0 y_1}+\cdots + \frac{1}{y_0 y_1 \cdots y_{n}}$. Prove that $$\lim_{n \rightarrow \infty} S_n = \frac{y_0 - \sqrt{y_0^2 - 4}...
1
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1answer
25 views

Construct Discrete Sequence in Complex space

Consider an arbitrary open subset $U \subset \mathbb{C}$. I intend to construct a sequence $(a_i)_{i \in \mathbb{N}}$ contained in $U$ with following two properties: for every rational point $q=q_R+...
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0answers
16 views

Big O and uniform estimates for eigenvalues of Dirichlet problem

I read the book "Inverse Spectral Theory" by J. Pöschel and E. Trubowitz. We have to prove the following theorem about location of eigenvalues for Dirichlet problem for $-y''+q(x)y=\lambda y$ (but it ...
1
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1answer
24 views

Is the projection onto the unit circle Sobolev?

Let $f(x,y)=\frac{x}{\sqrt{x^2+y^2}}$. Does $f \in W^{1,p}(B)$ for some $p \ge 1$, where $B$ is the open unit disk in $\mathbb{R}^2$? (I guess we can replace $B$ with a disk with arbitrarily small ...
0
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2answers
48 views

How to prove that $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\infty}f_\epsilon(x)g(x)dx=g(0)$ (Dirac delta function))

I'm currently studying the Dirac delta function using a textbook which unfortunately provides only partial solutions to its explanations. Why does $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\...
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1answer
15 views

Proving that the equivalence of paths is an equivalence relation.

The definition of equivalent paths is as follows : Two paths $f: [a,b] \rightarrow \mathbb{R^n} $ and $g: [c,d] \rightarrow \mathbb{R^n} $ are equivalent if there exist a $C^{1}$ bijection $\phi: [...
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0answers
40 views

Showing that $\lim_{n \to \infty}\frac{c_n}{4^n} = 0$

Here $c_n$ represents the Catalan numbers. This question is from an old exam paper with no solutions available. I have an approach to the problem but it feels very long-winded considering only a few ...
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0answers
29 views

How to find primitive function of [on hold]

$$\int\frac{x\sqrt{x+1}}{x-\sqrt{x-1}}$$
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2answers
35 views

Showing that 2 paths are not equivalent.

The definition of equivalent paths is as follows : Two paths $f: [a,b] \rightarrow \mathbb{R^n} $ and $g: [c,d] \rightarrow \mathbb{R^n} $ are equivalent if there exist a $C^{1}$ bijection $\phi: [...
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1answer
46 views

Vanishing of the integration along vertical line

2- Show that if $C$ is a vertical line segment $c \leq y \leq d,$ and if $F$ is a function of 2 variables defined on $C$, then $$\int_{C} F(x,y)dx = 0$$. I understand that the integration is ...
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1answer
23 views

maximum of the function $f(x) = \sum_{n\geq 1 } \sin n x /n^\alpha $

Here $\alpha >1 $. The function is defined as $$f(x) = \sum_{n\geq 1 } \sin n x /n^\alpha .$$ The domain is $(0, \pi)$. We know that if $\alpha = 1$, $f(x) = (\pi -x )/2$. If $\alpha >1$, ...
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1answer
40 views

Show that $\lim_{p\to0}\|x\|_p$ exists and determine its value (we also allow infinity as a limit).

Let $x \in \mathbb{R}^n$ and $$\|x\|_p= \left(\sum_{i=1}^n |x_i|^p\right)^ {1/p}\ $$ for $0< p < \infty $ and $\|x\|_\infty=\max_{i=1,...,n}|x_i|$. Show that $\lim_{p\to 0} \|x\|_p$ exists ...
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0answers
12 views

Subalgebras and dense

I'm trying to prove that $X=[0,\pi]$ and $A=\{\cos nx,\, \sin mx \mid m, n \in \mathbb{N} \cup \{0\}\}$ are dense subalgebras of $C(X).$ I know that I should prove that for two arbitrary elements in ...
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2answers
52 views

Rudin Principles of Mathematical Analysis Chapter 10, Exercise 8

I'm working on exercises of chapter 10 in Baby Rudin. I refer to R. Cooke's solutions manual to Baby Rudin while I'm solving those exercises.(https://minds.wisconsin.edu/handle/1793/67009) But I ...
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1answer
21 views

The integration of the norm of the derivative of 2 equivalent paths are equal.

Let $f: [a,b] \rightarrow \mathbb{R^n} $ and $g: [c,d] \rightarrow \mathbb{R^n} $ be 2 equivalent paths. Prove that $$\int_{a}^{b} ||f '(t)|| dt = \int_{c}^{d} ||g '(t)||dt. $$ The definition of ...
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2answers
63 views

Proving a Line-integration along a parametrized curve identitiy.

(this question were asked after studying line integrals) 1- Show that if $C$ is the graph of $y=f(x)$, $a \leq x \leq b,$ and if $F$ is a function of 2 variables defined on C, then $$\int_{C} F(x,y)...
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2answers
23 views

About $l^1$ norm

By duality and Hahn Banach theorem, we know that for $x\in \ell^1$, its norm can be computed as $$\|x\|_1=\sup_{\|\beta\|_\infty=1} \left|\sum_k x_k \beta_k\right|.$$ To obtain the norm, in that ...
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0answers
11 views

Differentiability by dense set

Actually this question comes from a hint of Excercise 1.1 of "Introduction to Ratner's theorems" by Dave Witte Morris. So we have to show that for any closed set $C$ of $\mathbb{T}^2$ (torus), there ...
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1answer
12 views

Constant for a Poincaré-Type Inequality

Given a bounded oben set $\Omega\subset\mathbb{R}^n$ and an arbitrary function $u\in C^2$. Is it possible to show $$ \|u\|_{L^2(\Omega)}^2 \leq C \|\nabla u\|^2_{L^2(\Omega)}, $$ for some $C\in\mathbb{...
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1answer
25 views

Book about summable families

after studying summable series, I know that there exists the concept of summable families. There is any book where I can find the proof of the basic properties of summable families? Thanks.
1
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1answer
42 views

$f,g \in [0,1] \times [0,1]$, $\int f - g \mathrm{d}x = 0$ and are monotonically increasing, then $\int |f-g| \mathrm{d}x \le \frac{1}{2}$

$f,g$ are monotonically increasing in $[0,1]$ and $0\le f , g \le 1$. $\int_0^1 f - g \mathrm{d}x = 0$. Prove that $$\int_0^1 |f - g|\mathrm{d}x \le \frac{1}{2}$$ In my previous question, $g(x) = x$....
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2answers
27 views

Constraints on $\alpha, \beta$, if $\alpha f(x)^2 + \beta f(x) + \gamma \equiv 0$

I am dealing with an expression of the form $$\alpha f(x)^2 + \beta f(x) + \gamma = 0,$$ where $f: \mathbb{R} \to \mathbb{R}$ is smooth and not identically zero, and $\alpha$, $\beta$, and $\gamma$ ...
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0answers
19 views

Determine whether sequence is convergent or divergent

I have the following sum $\sum_{i=1}^\infty \frac{(-1)^i * |a|^\frac{1}{i}}{i} $. For what values of a does this converge? Using alternating series test if a >= 1 It converges, but for a < 1, I am ...
1
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3answers
71 views

Limit as $x \to 0$ of $\big(x \int_x^1{\frac{f(t)}{\sin^2(t)}}\: dt\big)$

Suppose $f$ is continuous on $[-1,1]$ and differentiable on $(-1,1)$. Find: $$\lim_{x\rightarrow{0^{+}}}{\bigg(x\int_x^1{\frac{f(t)}{\sin^2(t)}}\: dt\bigg)}$$ I am trying to use l'hopital's rule ...
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0answers
28 views

Integral transform with reciprocal complex exponential functions?

I tried answering a question that ended up with an expression $$\mathcal F\left\{e^{\left(\frac{2\pi j} {t}\right)}\right\}$$ Now this function we know from famous identity is $$e^{ai} = \cos(a)+i\...
1
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1answer
26 views

Finding an example of exterior measure

I am finding a sequence of set $E_1\supset E_2 \supset ...$,and $m^*(E_k)<\infty $,satisfy$$m^*(\cap_{k=1}^{k=\infty}E_k)<\lim_{k\rightarrow \infty}m^*E_k$$ Attempt I think maybe I should use ...
0
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1answer
23 views

Exponential mapping in linear space

i am studying Functional Analysis. My teacher gives me a definition of exponential mapping Let $X$ is a banach space. $A \in L(X,X)$. We define $e^{A}=\displaystyle \sum_{n=0}^{\infty}\dfrac{A^n}{n!}...
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0answers
10 views

Geometric interpretation of Ellipticity Condition. [on hold]

A partial differntial operator $L$ is called (uniformly) elliptic if there exists a constant $M>0$ such that $\sum_{i,j=1}^{n} a_{ij}(y)x_i x_j\geq M {\lvert x\rvert}^2$ for all $y\in U$ and for ...
0
votes
1answer
27 views

Derivative of functions of several variables

From the definition of derivative of a function $f:\Bbb{R}^m \to\Bbb{R}^n$,how do I conclude that $f(x+h)-f(x)=f'(x)h+r(h)$ where $|r(h)|/h$ tends to $0$ as $h$ tends to $0$? I can't to do this. Sorry ...
6
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3answers
220 views

Will these geometric means always converge to $1/e$?

Let $p_n$ be the $n$-th prime and $F_n$ be the $n$-th Fibonacci number. We have $$ \lim_{n \to \infty}\frac{(p_1 p_2 \ldots p_n)^{1/n}}{p_n} = \lim_{n \to \infty}\frac{\{\log(F_3)\log(F_4)\ldots \...
1
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0answers
28 views

small parameter

There is differential equation $\phi''+ k^2 \sin \phi + \eta S \phi'=0, \phi(0)=\alpha_0,\phi'(0)=0$, where $\eta$ is small. $\phi=\phi_0+\eta \phi_1+...$ I use a $sin(\phi)\;$Taylor series around $\...
0
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2answers
54 views

Prove that a bounded a bounded set in $\mathbb{R^2}$ with a finite number of accumulation points has content 0.

Prove that a bounded set in $\mathbb{R^2}$ with a finite number of accumulation points has (Jordan ) content 0. My thoughts: First the definition of a set has (Jordan) content 0 given in the book is ...