Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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19
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0answers
223 views

Strengthening the intermediate value theorem to an “intermediate component theorem”

Let $f$ be a continuous function on a closed Euclidean ball (in dimension $\ge 2$) that is negative in the center of the ball and positive on its boundary. On any path from the center of the ball to ...
18
votes
0answers
297 views

Can we find this infinite root in term of elementary function?

Let $f(x)=\left(x+f(x+1)\right)^\frac{1}{x}$. What is the value of $f(2)$ ? More precisely, how to find the value of $$\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}}~?$$ Thank you.
11
votes
0answers
334 views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that ...
10
votes
0answers
379 views

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $

Background: On my quest to solve difficult integrals, I chanced upon this site: http://www.durofy.com/5-most-beautiful-questions-from-integral-calculus/ Good problems for me, (novice), although I ...
10
votes
0answers
64 views

How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a ...
10
votes
0answers
233 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\bigl(\frac{1}{e^n}\bigr)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
10
votes
0answers
387 views

Integral $\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}\mathrm dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set $a=0$ we get a similar integral given by $$ ...
10
votes
0answers
444 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
9
votes
0answers
339 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
9
votes
0answers
92 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
9
votes
0answers
301 views

Evaluate $\int_0^{\infty} \frac{1-e^{-ax}}{x e^x} dx$

I found two different approaches, both is giving the same answer. Fubini: $$ \begin{align} \int_0^{\infty} \frac{1-e^{-ax}}{x e^x} \,dx &= \int_0^{\infty} e^{-x} \int_0^a e^{-xy} \,dy\, dx \\ ...
9
votes
0answers
291 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
8
votes
0answers
99 views

Inequality involving exponential partial sums

Consider the exponential partial sums $E_n(x) = \sum_{i=0}^n \frac{x^i}{i!}$. I want to prove that for all $x \ge 0$: $$2 \frac {E_{n-1}(x)} {E_n(x)} \ge \frac {E_{n}(x)} {E_{n+1}(x)} + \frac ...
8
votes
0answers
80 views

Does $\sum_n |\sin n|^{cn^2}$ converge?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don't even converge, for any $c > 0$, so the series ...
8
votes
0answers
115 views

Integral Asymptotics for inhomogenous phase

I'm looking for asymptotics for an integral of the form: $$F(n):=\int_{1/2-i\infty}^{1/2+i\infty} e^{\phi(n,z)}dz$$ where $\phi(n,z)=(n-n^3)\log(1-z)+n^2\log(1+z)-n\log(z)$. One can solve for the ...
8
votes
0answers
324 views

Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
8
votes
0answers
76 views

Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$

Now since the sum $$ \sum_{n=0}^\infty \frac{x^n}{n!},\quad x\in\Bbb R, $$ does have some relatively nice properties, is the same true for its analogues integral? If we take the gamma function to be a ...
8
votes
0answers
139 views

Limits of series, proof of the convergence of two sequences

I have two sequences $x_i$ and $y_i$ defined by their expressions : $$x_i-x_{i+1}=y_i-y_{i+1}=\sqrt{x_{i+1}y_{i+1}}$$ I have to prove that $xy(x-y)=0$. I tried this : I have ...
8
votes
0answers
244 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
8
votes
0answers
400 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
8
votes
0answers
295 views

Laplace-Beltrami Operator for Euclidean Space

Consider the space $\mathbb{R}^n$ and let $x_1,\ldots, x_n$ be the coordinates. Fix the orientation $dx_1\wedge dx_2\ldots\wedge dx_n$. Let $E^p$ denote the space of smooth $p$ forms and let $d$ ...
8
votes
0answers
207 views

Invariant functions on the space of finite sequences of reals

Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation. 1. For each $\mathbf s\in S$ we ...
7
votes
0answers
47 views

If $f:I\to \mathbb{R}$ is differentiable and $f'$ is monotone nondecreasing then $f$ is convex.

Let $I \subseteq \Bbb{R}$ be an open interval. If $f:I\to \mathbb{R}$ is differentiable and $f'$ is monotone nondecreasing then $f$ is convex. I would like a hint to solve this exercise. I ...
7
votes
0answers
68 views

one-sided differentiability

Well known theorem: If $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable and $f'(x)=0$ for all $x$, then $f$ is constant. The assumption of differentiability can be weakened to continuity and ...
7
votes
0answers
142 views

Evaluating $\int_0^{\infty} \log(\sin^2(x))\left(1-x\operatorname{arccot}(x)\right) \ dx$

One of the ways to compute the integral $$\int_0^{\infty} \log(\sin^2(x))\left(1-x\operatorname{arccot}(x)\right) \ ...
7
votes
0answers
149 views

Closed form of $\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$

Find the closed form of $$\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$$ I know the closed form for smaller powers like $2, 3$ exists, but I'm not sure if there is a closed form for this ...
7
votes
0answers
85 views

Dimension score for metric spaces $\mathbb{R}^n$

Given any metric space $(X,d)$, define its score $S(X)$ to be the smallest value of $k$ such that for every $x\in X$ and $r>0$, the ball $B(x,r)$ is covered by at most $2^k$ balls of radius $r/2$. ...
7
votes
0answers
198 views

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
7
votes
0answers
130 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
7
votes
0answers
86 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really ...
7
votes
0answers
120 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
7
votes
0answers
324 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
7
votes
0answers
260 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...
7
votes
0answers
224 views

Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$

I am trying to solve a qualifying exam problem. I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup ...
7
votes
0answers
475 views

Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
7
votes
0answers
93 views

Compute $\int_{0}^{+\infty} e^{-y}y^a\mathrm{d} y$

Does the following integral have a finite value? How to compute it? $$\int_{0}^{+\infty} e^{-x^k}\mathrm{d} x$$ where $k$ is given and $0<k<1$. By substituting $x^k=y$ we may obtain an ...
6
votes
0answers
95 views

Find the closed form of $\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$

I was thinking of the generalization of the problem here, that is $$\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$$ Maybe you recommend me some tools? ...
6
votes
0answers
87 views

Borel measurability is a local property

I am looking at Exercise 5.2 (page 44) in "Real Analysis for Graduate Students" by Richard Bass. Let $f:(0, 1)\to \mathbb{R}$ be a function such that for every $x\in (0, 1)$, there exist ...
6
votes
0answers
99 views

The dual of the Banach space $C(\Omega)$

It is well-known that the dual of the Banach space $C([0,1])$, i.e. the space of all continuous functions on the interval, is the space of all functions of bounded variation on the interval, ...
6
votes
0answers
128 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
6
votes
0answers
121 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
6
votes
0answers
48 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
6
votes
0answers
86 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that ...
6
votes
0answers
166 views

The elementary methods to compute $\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\quad;\quad\text{for}\, \alpha>0$

How to compute the following integral using elementary methods (high school methods). \begin{equation}\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\qquad;\qquad\text{for}\, ...
6
votes
0answers
119 views

Theorem $2.14$ page $40, 41$ in Rudin - Real and Complex Analysis

Can anyone tell me the signification of Theorem $2.14$ (The Riesz Representation Theorem in locally compact Hausdorff spaces), page $40, 41$ in Rudin - Real and Complex Analysis? And some applications ...
6
votes
0answers
274 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
6
votes
0answers
91 views

Asymptotic property of integral involving Bessel function.

Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, ...
6
votes
0answers
705 views

Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$

As a homework question, we got asked the following: Construct a function $f:[0,1] \rightarrow \mathbb{R}$ which is not the pointwise limit of any sequence of continuous functions Thinking about ...
6
votes
0answers
158 views

Is there a subsequence of $a_n = n \sin(n)$ which tends to $0$?

I know there is such a subsequence for $b_n = \sin(n)$. What about $a_n = n\sin(n)$?
6
votes
0answers
91 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...