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
87 views

$h^{n} = f$, $h$ and $f$ entire functions

I found this exercise. Let $f$ be an entire function and $n$ a positive integer. Show that there exists an entire function $h$ such that $h^{n} = f$ if and only if the orders of the zeros of $f$ are ...
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1answer
374 views

$\inf{(A+B)}=\inf{A}+\inf{B}$: An $\epsilon$ Proof

$$\text{COMPARATIVE EXAMPLE}$$ So I've been told that in order to show that $$\sup{(A+B)}=\sup{A}+\sup{B}$$ for non-empty and bounded above sets $A,B\subseteq\mathbb{R}$ one must show that ...
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2answers
70 views

What can the function f look like?

Im reading Carothers' Real Analysis, 1ed talking about Continuity and Category. Here is a corollary, What puzzling here is, for example, f is continuous at all points of $\mathbb Q$ and then it ...
4
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2answers
135 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
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2answers
89 views

There are $u$ in $W^{1,p}(D)$ and a weakly converging subsequence $\left\{ u_{m_{k}}\right\} $ to $u$.

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$, $p \in (1,\infty)$, and {$u_m$} be a bounded sequence in $W^{1,p}(D)$. Then there are $u$ in $W^{1,p}(D)$ and a subsequence ...
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1answer
31 views

Prove the following subspace is closed in $L^2(\mathbb{R})$

Let $V_j=\{f\in L^2(\mathbb{R}): f$ is constant on $[\frac{n}{2^j},\frac{n+1}{2^{j}}) $for all $ n \in\mathbb{Z}\}$ , $j\in\mathbb{Z}$ be a sub set of $L^2(\mathbb{R}).$ Prove that $V_j$ is an closed ...
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1answer
75 views

uniform convergence of a sequence of linear transformations on a compact subset

Let $\{T_n\}$ be a sequence of linear transformations from $\mathbb{R}^k$ to $\mathbb{R}^m$. If $\{T_n\}$ converges pointwise to $T$, then $T$ is a linear transformation; the convergence is uniform ...
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2answers
1k views

Laplacian identity.

Consider $f$ and $g$ smooth functions. How to prove the following identity: $$\Delta\left(\frac{f}{g}\right)=\frac{1}{g}\Delta f-\frac{2}{g}\nabla\left(\frac{f}{g}\right).\nabla g-\frac{f}{g^2}\Delta ...
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2answers
2k views

Prove $x^n$ is not uniformly convergent

This question pertains to the sequence of functions $f_n(x)=x^n$ on the interval $[0,1]$. It can be shown this sequence of functions ${f_n}$ converges point-wise to the limit $f$ where $f$ is defined ...
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0answers
78 views

Improper Riemann integral and imaginary exponential of real polynomials

Let $P(x_1,\cdots,x_n)$ be a real polynomial of degree $\geq 2$. What are the conditions on $P$ so that $$ I_P:=\int_{\mathbb{R}^n} e^{iP(x)} dx $$ exists as an improper Riemann integral ? Already ...
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0answers
118 views

Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ a.e. on D.

I want to prove this result: Let $D$ be an open subset of $\mathbb{R}^n$, $p \in[1,\infty)$ and $f$ be in $L^p(D)$. Assume $\int_{D}fgdx=0\forall g\in C_{c}^{\infty}\left(D\right)$. Then $f = 0$ ...
3
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1answer
72 views

Limit of an integral with changing domain

I'm trying to figure out the limit : $$\lim_{n\rightarrow \infty} \int_{0}^{\sqrt{n}} \left(1-\frac{x^2}{n}\right)^{\!n}dx$$ Now the problem is that as $n$ rises so does the range of integration, ...
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0answers
136 views

Derivative of the norm of a second order tensor w.r.t. the tensor

I came across this derivative in a paper: $\frac{\partial{\|{A}\|}}{\partial A} = \frac{A}{\|A\|}$ where A is a rank 2 tensor. And I wonder what is the definition of this norm. I believe it is not a ...
0
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1answer
43 views

Continuous extension to extended reals.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous periodic function such that $f$ extends to a continuous function from $\overline{\mathbb{R}}$ to $\overline{\mathbb{R}}$. Prove that $f$ is ...
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1answer
88 views

Showing a function is limited

This is something about which I'm quite confident, but I would really like to be sure Let $f(x)$ be a continuos function in $\mathbb{R}$ Then if $\displaystyle \lim_{x \to \pm\infty} f(x) = l \in ...
4
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1answer
172 views

Integral $\int_{-1}^{1}\frac{\sqrt{(1-x^{2})}}{1+x^{2}}dx$ [duplicate]

Consider $$\int_{-1}^{1}\frac{\sqrt{(1-x^{2})}}{1+x^{2}}dx$$ I have a problem with this integral; the method I know consists in calculating the complex integral of $$f(z) = \left( \frac{z-1}{z+1} ...
0
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1answer
295 views

Notation Question (Meaning of double inclusion symbols)

What does the notation $\subset \subset$ mean? In my class notes, our prof writes $\Omega \subset \subset \mathbb{R}^{n}$ to mean that "$\Omega$ is a convex subset of $\mathbb{R}^{n}$". Is that all ...
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1answer
94 views

There exist such f, f'

My question is: There exist a function $f = f(x)$ that satisfy $$\lim_{x \to +\infty} f(x) = +\infty$$ $$\lim_{x \to -\infty} f(x) = +\infty$$ $$\lim_{x \to +\infty} f'(x) = +\infty$$ $$\lim_{x \to ...
1
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1answer
148 views

Relation between convergence in distribution and weak convergence

If $(X_n: n\in \mathbb{N}), X$ are a sequence of random variables in $\mathbb{R}$, I wish to show that $X_n \to X$ weakly if and only if $X_n \to X$ in distribution. By 'converging weakly' I mean that ...
2
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2answers
203 views

Decimal Digits and Unique Real Numbers

I understand that every real number has an infinite decimal expansion. How can I use the Axiom of Completeness to prove that every string of decimal digits corresponds to a unique real number alpha? ...
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1answer
801 views

Finding the interval where a solution is certain to exist for the equation $y' + (\tan t)y = \sin t$

Given the following problem: Determine (without solving the problem) an interval in which the solution is certain to exist for the initial value problem $y' + (\tan t)y = \sin t, \space y(2\pi) = ...
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1answer
25 views

Modification on a corollary

captured from Chapter11 of Carothers' Real Analysis, 1ed and X denotes a compact metric space. Can I claim that uniformly convergent is not necessary here cos every convergent sequence in C(X) is ...
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1answer
32 views

Line integral and dependenece on direction

I have been doing some mistake for quite a while now and I don't see it. Therefore, I wanted to ask you whether anybody here could tell me what I have been doing wrong: $\gamma:[0,1] \mapsto ...
0
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1answer
183 views

Is a totally bounded set necessarily closed?

If A is a totally bounded set in metric space X, then A is closed in X? If not, can you show me some examples?
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1answer
45 views

a Problem about Special Sequence

I am doing homework. And this question may related to #Proving statement about sequences. However, I want to consider the limit of it. Let we assume $a_1=a,b_1=b,b>a>0$. Then we consider the ...
0
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1answer
99 views

a Problem about Sequence [duplicate]

Let $a_1$ be an integer. Then we assume $$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$ Now we prove that for any ...
2
votes
1answer
227 views

Conditions implying that image of $f$ contains the unit disc

I'm stuck with this problem from Stein-Shakarchi: Let $f$ be non-constant and holomorphic in an open set containing the closed unit disc. a) Show that if $|f(z)| = 1$ whenever $|z| = 1$, then the ...
31
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2answers
960 views

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$, such that $$ f(x)+f(x^2)=x. $$ My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all ...
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2answers
60 views

Some contradictions in C(X)

I'm reading Carothers' Real Analysis, 1ed. Here are two functions from C[0,1] used in his book, See, both of them ∈ C[0,1] while function(b) is not bounded actually and function(c) does not ...
3
votes
1answer
384 views

Is Morera's theorem the inverse theorem of Goursat's theorem?

While I'm reading Complex Analysis by Elias M.Stein, I found that there must be some relations between Goursat's theorem and Morera's theorem. According to Stein, the 2 theorems are as following: ...
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1answer
94 views

uniform continuity and which of the following statements are true??(NBHM-$2014$)

Let $g_n(x)=n[f(x+\frac{1}{n})-f(x)]$, where $f: R\to R$ is a continuous function . Which of the following are true? a. If $f(x)=x^3$, then $g_n\to f'$ uniformly on $R$ as $n\to \infty$. b. If ...
1
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1answer
48 views

to show a function is bounded when domain is bounded

How to show that $f$ : $S$ -> $R$ is uniformly continuous and $S \subset R $ is bounded then $f$ must be bounded. I have tried using the theorem which is states the three statements to be ...
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1answer
1k views

Brouwer's Fixed Point theorem proof for 2-dimension

I am trying to find a elementary proof of the Brouwer's fixed point theorem only using basics of point set topology and real analysis. In the one of the textbooks I read, they were proving Brouwer's ...
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1answer
99 views

A basic question about function space

Im reading Carothers' Real Analysis, 1ed. Actually, Carothers has begun his talk of function space since chapter9. However, I haven't found any definition about function space. Here is a definition ...
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1answer
59 views

Open Nested Interval Question [duplicate]

Find a family $\{ I_n \}$ of open nested intervals such that no two $I_n$ are equal and the intersection is equal to $\left[-2,2\right]$.
4
votes
2answers
210 views

A first-order non-linear ordinary differential equation containing various squares

The Equation: Find all differentiable functions $f: I \rightarrow \mathbb{R}$ satisfies: $$\big(\,f(x)-x\,f'(x)\big)^2 = \big(\,f'(x)\big)^2 + 1 \; \; \; \; \; \text{for all}\,\,\, x \in I,$$ where ...
0
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1answer
310 views

Zeros of $f_{\epsilon}(z) = f(z) + \epsilon g(z)$ with $f$ and $g$ holomorphic

I'm stuck with this problem from Stein-Shakarchi: Suppose $f$ and $g$ are holomorphic in a region containing the disc $|z| \leq 1 $. Suppose that $f$ has a simple zero at $z = 0$ and vanishes nowhere ...
0
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1answer
60 views

What's the difference between ε-dense and dense?

Im reading Carothers' Real Analysis, 1ed. Here is the definition of dense, and ε-dense is claimed in topic of total boundness, . What's the difference between ε-dense and dense?
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1answer
59 views

Family of Closed/Open Nested Intervals

Find a family $\{I_n\}$ of closed nested intervals, such that no two $I_n$'s are equal and their intersection is $[-2,2]$. An answer for the same question except for dealing with open nested ...
3
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1answer
55 views

To what extent can one imitate holomorphic functions in higher dimensions?

Let $\Omega\subseteq \mathbb{C}$ be open. Let $f:\Omega \rightarrow \mathbb{C}$ be holomorphic. The last statement about $f$ can be restated as: For every $z_0\in \Omega$, there exists $w\in ...
2
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3answers
121 views

Product of distances from a point in $\mathbb{C}$, Stein-Shakarchi

I 'm trying to do this exercise from Stein - Shakarchi. Let $w_{1} \ldots w_{n}$ be points on the unit circle in $\mathbb{C}$. Prove that there exist a point $z$ on the unit circle such that the ...
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2answers
374 views

Periodic Function with no Minimum is Constant

Let $f: \mathbb R \rightarrow \mathbb R$ be a periodic function, which means that there is some positive $p$ such that $f(x)=f(x+p)$ for all $x$. Is it the case that, if there is no minimum such $p$, ...
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2answers
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How prove this function $f(x)=x!-x^n$ is injective

Question: For any positive integer $n$ such $n\neq 2^m-1, n\ge 2$, and function $f$ defined by $$f(x)=x!-x^n$$ show that : $f:N\to Z$ is injective. My idea: maybe for $x\neq y$ with $x,y\in N^{+}$, ...
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1answer
43 views

Convex function in its interior

Let $f$ be a convex function on an open subset of $R^{n}$. How to prove $f$ is continuous in the interior of its domain. For $n=1$, let $f$ be convex on the set $(a,b)$ with $a<s<t<u<b$ ...
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1answer
372 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
0
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1answer
218 views

Axiom of Completeness for set of integers

If $A$ is a subset of the integers $\mathbb{Z}$, and is bounded above, then A has a supremum $\alpha$ that is an element of the integers $\mathbb{Z}$. Is this statement true?
0
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1answer
62 views

directional derivatives at (0,0) vanish

Is the statment that all directional dervatives vanish at (0,0) really true, it seems to me the last equation states the opposite. The example is from: Mathematical Analysis: An Introduction to ...
3
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2answers
87 views

when is this statement is true?

Let $a_i>0$ be a sequence in $\mathbb{R}$. It's well known that: $\sum\limits_{i=0}^{n}a_i\to a \ $(as $n\to\infty)\Longrightarrow a_i\to0$ (as $n\to\infty$) My question is when is the following ...
2
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1answer
48 views

$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$ is antiholomorphic

I 've encountered this fact: if $z \in D(0,1) $ and $f$ is continous on $\partial D(0,1) $ then $$g(z) = \int_{0}^{2\pi}f(e^{i\theta})\frac{e^{-i\theta}}{e^{-i\theta}-\bar{z}}d\theta$$ is ...
0
votes
1answer
43 views

Proof of generalization of a particular limit converging to $e^{\frac{1}{(p-1)^2}}$

I was reading a very old and long article on logarithms in a library it has pages turned yellow and had one pages titled - Tricky problems I managed to solve 5 out of the 6 but I couldn't do this 6th ...