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).

38,482 questions
Filter by
Sorted by
Tagged with
0 votes
0 answers
27 views

How can I rewrite this sets using measure theory?

Let $(\Omega, F, \mu)$ be a measure space. Let $f_n:\Omega\rightarrow \Bbb{R}$ be measurable functions. Let $$B:=\{x\in \Omega: \lim_{n\rightarrow \infty} f_n(x)~~\text{does not exists}\}$$I need to ...
• 1,625
-2 votes
0 answers
37 views

Of a Controversy concerning the Order of Magnitude of Functions

Some time ago I started a chapter in my Calculus/Intro-Analysis textbook that had for its subject the notion of Order of Magnitude of functions tending to 0 or increasing beyond all bounds by absolute ...
• 433
2 votes
2 answers
65 views

Using LMVT ( Lagrange's Mean Value Theorem) prove the following inequality

I want to prove the following inequality for $x > 0$ $$0 < \frac{1}{x} \log \frac{e^x - 1}{x} < 1$$ I think I can do this problem, but I just don't know what should be the function that is ...
0 votes
1 answer
26 views

Does Caretheodory's extension theorem of premeasures on sigma-finite measure space give us a complete measure?

From Folland's analysis textbook, we constructed Lebesgue-Measure using the function $F(x)=x$ and defined a pre-measure on intervals of the form $(a,b]$ and extended using Caretheodory's extension ...
• 3,191
0 votes
0 answers
19 views

Richardson's Extrapolation with a data tabel

I need to find the value of f'(1) using richardson's extrapolation formula and I have been given a set of data in a table to be used in the calculations. Based on the data, I assume that the step size ...
0 votes
2 answers
37 views

• 5,238
0 votes
1 answer
26 views

one-dimensional Gelfand problem

For $t_0>0$, we define $$I(t_{0})=\int_{0}^{t_{0}} \frac{dt}{\sqrt {e^{t_{0}}-e^t}},$$ how to proof that $I(\cdot)$ takes values in some bounded interval and achieves its maximum at a unique point.
0 votes
4 answers
84 views

Proof verification of the convergence question: If $\frac{1}{n}a_n\to0$, then $\frac{1}{n}\max\{a_1,\ldots,a_n\}\to0$.

I don't know how to use latex that well, so I'll upload the imagines of the question and the way I tried to prove it. I am unsure if my proof is correct. If it's not correct, then could you explain ...
• 11
2 votes
1 answer
67 views

existence of a solution of a differential equation

we have a continuous function $\omega:\mathbb{R}_+\rightarrow\mathbb{R}_+$ such that $$p'(t)=\omega(p(t))\\ p(0)=0$$ has only the trivial solution $p=0$ on $J=[0,b]$. Now the author ...
• 43
0 votes
1 answer
54 views

Cartan's proof that there is a smallest positive zero of cosine function

In section 3, paragraph 3 of Chapter 1 of his "Elementary theory of analytic functions of one or several complex variables", Cartan wants to prove that $\cos y$ vanishes for a certain value ...
• 1
2 votes
2 answers
65 views

Are these two limits equal?

The problem basically goes as follows: Let $x_i$ be a sequence of real numbers. Let $f : \mathbf{N} \to \mathbf{N}$ be a bijection. And finally, let $h : \mathbf{R} \to \mathbf{R}$ be some function. I ...
• 73
0 votes
0 answers
35 views

Some question of upper and lower limits in a paper.

I have read a paper and cofused some notions and some technical details. First, I state some definitions in this paper. Let $\mathcal{P}_\mathbb{N}$ and $\mathcal{P}_n$ be respectively the power sets ...
• 49
2 votes
1 answer
49 views

• 53
0 votes
0 answers
44 views

How can you prove that $B(x,r)\subseteq B(0,s)$ where $0<r<s$?

I'm approaching the problem in the following way: If $B(x,r)=\{y\in X:|x-y|<r\}$ and $B(0,s)=\{y\in X:|y|<s\}$ then it must be the case that $y\in (x-r,x+r)$ for the first ball and $y\in (-s,s)$ ...
0 votes
0 answers
13 views

• 3,191
0 votes
1 answer
60 views

Finding an Upper bound for modulus of a function

Consider the integral $$f(s)=\int_{0}^{\infty}\frac{\left(\frac{1}{2}+ix\right)^{1-s}}{\cosh ^2(\pi x)} dx$$ where $s=\sigma+it$ and $\sigma,t\in\mathbb{R}$. Find a Big-O upper bound for the above ...
4 votes
0 answers
36 views

Can differentiability classes be extended to negative values?

A function $f$ is said to be of differentiability class $C^k$ if its first $k$ derivatives are continuous. It has the property that $m>n \implies C^m \subset C^n$ and that all $C^k$s are algebras. ...
• 551
2 votes
3 answers
62 views

Calculate the area of the surface $x^2 + y^2 = 1 + z^2$ as $z \in [- \sqrt 3, \sqrt 3]$

The question states the following: Calculate the area of the surface $x^2 + y^2 = 1 + z^2$ as $z \in [- \sqrt 3, \sqrt 3]$ My attempt In order to solve this question, the first thing I think about is ...
• 108
-1 votes
0 answers
19 views

How do I show that show that lim (n^2/n!) = 0? [duplicate]

since this is the same as nn/(n-1)!n = n/(n-1)!, i can also prove that this goes to zero right? But I don't have any idea on how to write the prove.
2 votes
0 answers
30 views

What is the equivalent condition for a smooth function on $\mathbb{R}^n$ to be well-defined as a smooth function on $S^{n}$?

The question is as in the title. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a smooth and bounded function. What would be the necessary and sufficient condition for this $f$ to be extended to a smooth ...
• 5,442
1 vote
0 answers
50 views

0 votes
1 answer
38 views

Questions regarding the Residue theorem (complex analysis)

I am trying to figure out how to calculate the residue of a function f(z) at z=$\infty$. My best guess is to define $g(z)=f(\frac{1}{z})$ then find the Laurent series for g(z) at z=0 and then find the ...
• 11
2 votes
2 answers
136 views

• 437
0 votes
0 answers
10 views

Why is the $D_{KL}(P||Q)\neq\infty$ as $P$ and $Q$ on the same support?

Define the Kullback-Lieibler (KL) divergence for two probability measures $P$ and $Q$ : $$D_{KL}(P||Q):=\int p(x)\log (\frac{p(x)}{q(x)})dx$$ I try to convince myself that the KL divergence for the ...
• 23
-5 votes
0 answers
43 views

Finding $\lim \limits _{(x,y)\to (0,0)}\frac{x^4-y^4}{x^2+y^2}$ [closed]

Determine whether the following limit exist, if so find the value:$$\lim \limits _{(x,y)\to (0,0)}\frac{x^4-y^4}{x^2+y^2}.$$
3 votes
0 answers
111 views
+100

$\lim_{r\to\infty} \int \cos^2(rz + t)\, d\pi_{\#}\mu(z) = \frac12$ for all $t\in \Bbb R$

This question stems from the identity listed as Equation $(6.6)$ in this paper on Pg. $11$. We want to show $$\color{blue}{\lim_{r\to\infty} \int \cos^2(rz + t)\, d\pi_{\#}\mu(z) = \frac12} \tag{6.6}$$...
• 10.5k
1 vote
0 answers
47 views

Proof in set theory with infinite indices

Prove that $\bigcap_{k=1}^{\infty}\{ \bigcup_{n=k}^{\infty} E_n\}=\{ x: x \in E_n$ for an infinite number of indices $n \}$ Can't solve that, just trying to begin with contradiction: $x \in E_n$ for ...