For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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67
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0answers
2k views

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
14
votes
0answers
299 views

A closed form of $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$?

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ \sum_{k=1}^\...
14
votes
0answers
261 views

$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^...
13
votes
0answers
103 views

Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,...
13
votes
0answers
570 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 ...
12
votes
0answers
162 views

Regularity of PDEs, general question (Example: Kolmogorov equation)

Let us consider a differential operator $L$ and the PDE $$Lu=0$$ with some boundary conditions (assume Neumann conditions). For example, Kolmogorov's equation $$(1) \quad Lu(t,x) = \partial_t u(t,x) + ...
12
votes
0answers
195 views

Dilogarithm identity containing the tribonacci constant

The motivation of this question is the brilliant conjecture by @Tito Piezas III. In $(4)$ of his question the equation seems to be true for all $n > 1$ real numbers. The case $n=2$ leads us to a ...
12
votes
0answers
323 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 variant....
11
votes
0answers
228 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
11
votes
0answers
208 views

A cotangent series related to the zeta function

$$\sin x = x\prod_{n=1}^\infty \left[1-\frac{x^2}{n^2\pi^2}\right]$$ If you apply $\log$ to both sides and derivate: $$\cot x = \frac{1}{x} - \sum_{n=1}^\infty \left[\frac{2x}{n^2\pi^2} \frac{1}{1-\...
10
votes
0answers
73 views

Evaluation of $\int\frac{(1+x^2)(2+x^2)}{(x\cos x+\sin x)^4}dx$

Evaluation of $$\int\frac{(1+x^2)(2+x^2)}{(x\cos x+\sin x)^4}dx$$ $\bf{My\; Try::}$ We can write $$x\cos x+\sin x= \sqrt{1+x^2}\left\{\frac{x}{\sqrt{1+x^2}}\cdot \cos x+\frac{1}{\sqrt{1+x^2}}\cdot \...
10
votes
0answers
120 views

Proof of the relation $\int^1_0 \frac{\log^n x}{1-x}dx=(-1)^n~ n!~ \zeta(n+1)$

I had the thought that by introducing some parameters into simple integrals and taking derivatives we can get exact values for infinitely many 'complicated' integrals. $$\int_0^1 x^a dx = \frac{1}{a+...
10
votes
0answers
134 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
10
votes
0answers
210 views

Find the limit of $ \lim_{n\to\infty}\frac{n}{\ln{n}}\left(\frac{1}{p+1}-na_{n}^{p+1}\right) $

Problem:Let postive real sequence$\{a_{n}\}$ satisfy $\displaystyle\lim_{n\to\infty}a_{n}\left(\sum_{i=1}^{n}a_{i}^{p}\right)=1$,where $p>-1$,Find the limit. $$ \lim_{n\to\infty}\frac{n}{\ln{n}}\...
10
votes
0answers
220 views

An integration to first order

I am having some trouble evaluating an integral -- involving taking an approximation. It would be great if someone could help me. I wish to evaluate $$\int_0^\pi {\cos\theta\cos \left[\omega t-{\...
9
votes
0answers
100 views

Suppose $f$ is continuous on $[0,2]$ and $f(0) = f(2)$. For which $a\in(0,2)$ must there exist $x,y\in[0,2]$ so that $|y − x| = a$ and $f(x) = f(y)$?

Suppose $f$ is continuous on $[0,2]$ and $f(0) = f(2)$. For which $a\in(0,2)$ must there exist $x,y\in[0,2]$ so that $\lvert y − x\rvert = a$ and $f(x) = f(y)$ I'm really unsure how to approach ...
9
votes
0answers
109 views

Floor function and convergence of the sequence

Sequence $\{a(n)\}$ of real numbers is such that $\forall\space\lambda\in(1,2)$ sequence $a(\lfloor{\lambda}^n\rfloor)$ has a finite limit. Does it follow that $\{a(n)\}$ is convergent?
9
votes
0answers
114 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq 0}...
9
votes
0answers
227 views

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any ...
9
votes
0answers
176 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider $f(x)=...
8
votes
0answers
143 views

prove uniqueness from orthogonality relation

The problem: we have two functions $f(x), g(x)\in C^{1}[-\pi, \pi]$, and we know that \begin{align} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \cos\left(ky\right) \sin\left(k\left\lvert y-z\right\rvert\...
8
votes
0answers
212 views

Brezis Exercise 3.27 Extension.

Let $E$ be a separable Banach space with norm $\|\cdot\|$. The dual norm on $E^*$ is also denoted $\|\cdot\|$. Let $(a_n) \subset B_E$ be a dense subset of $B_E$ with respect to the strong ...
8
votes
0answers
171 views

How to evaluate the integral $\int_0^\infty \frac{x^{a-1}}{1+bx^a} e^{-x} dx$

How to evaluate this integral? \begin{equation} \int_0^\infty \frac{x^{a-1}}{1+bx^a} e^{-x} dx \end{equation} I think it will use a gamma function or a exponential integral. I really need an ...
8
votes
0answers
159 views

An Additional Rule for Calculus

Background The rules for differentiating elementary functions (arithmetic, exponential, trigonometric, etc.) together with the chain rule for differentiating compositions of functions are often ...
8
votes
0answers
177 views

A difficult contest question from the former Soviet Union

Let $(a_n)$ be a positive sequence such that $\varlimsup\limits_{n\to\infty} a_n^{1/n}=1$ and $\varliminf\limits_{n\to\infty} a_n^{1/n}<1$. Prove there exists a subsequence $(a_{n_i})$ such that $...
8
votes
0answers
198 views

Modern notational alternatives for the indefinite integral?

I like the Leibniz notation, and I think the reason it's survived for over 300 years and continued to be almost the only game in town is that in many respects it's a miracle of design. Nevertheless it'...
8
votes
0answers
410 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
8
votes
0answers
386 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 \...
8
votes
0answers
4k views

How to find probability distribution function given the Moment Generating Function

After searching, I found two questions like mine, but didn't see my answer to my question. Finding a probability distribution given the moment generating function Finding probability using moment-...
7
votes
0answers
132 views

prove that $\displaystyle \sin (\tan x)\geq x\;\forall x\in \left[0,\frac{\pi}{4}\right]$

Using the relation $2(1-\cos x)<x^2,x\neq 0$ or otherwise, prove that $$ \sin (\tan x)\geq x\;\forall x\in \left[0,\frac{\pi}{4}\right] $$ My Attempt: Let $f(x) = \sin (\tan x)-x$. Then $f'(x) ...
7
votes
0answers
99 views

Infinite Integration in Limits of Integration

Given the following: $$ u_0 = \int \limits_{ 0 } ^{ 1 } x \, dx , \:\:\: u_1 = \int \limits^{ \int \limits_{ 1/2 } ^{ 1 } x \, dx } _{ \int \limits_{ 0 } ^{ 1/2 } x \, dx } x \,dx , \:\:\: u_2 = \int \...
7
votes
0answers
119 views

Bear of an integral

I have a pretty ferocious integral to solve, and would be over the moon if I were able to get some sort of analytic expression / insight for it. $$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, \...
7
votes
0answers
243 views

An alternative proof of Cauchy's Mean Value Theorem

Let's focus on the following version of Cauchy's Mean Value Theorem: Cauchy's Mean Value Theorem: Let $f, g$ be functions defined on closed interval $[a, b]$ such that 1) Both $f, g$ are ...
7
votes
0answers
91 views

Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...
7
votes
0answers
179 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot \frac{1}{1+...
7
votes
0answers
101 views

Can these two indefinite integrals be evaluated in closed form?

I'm wondering whether any of these two indefinite integrals $$\int \frac{1}{\sqrt{1+\alpha \sinh(x)^{-4/3}}}dx$$ $$\int \frac{\sinh(x)^{-4/3}}{\sqrt{1+\alpha\sinh(x)^{-4/3}}}dx$$ can be evaluated in ...
7
votes
0answers
451 views

Integration of product of functions(Special form)

Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any ...
7
votes
0answers
144 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
182 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The Euler-...
7
votes
0answers
158 views

How to solve this integral equation?

Solve this integral equation: $$ {{\rm e}^{{\rm i}k\,\sqrt{\vphantom{\Large A}\,r^{2} + z^{2}\,}\,} \over \sqrt{\vphantom{\large A}r^{2} + z^{2}\,}} = \int_0^{\infty}{\rm K}_{0}\left(\lambda r\right) \...
7
votes
0answers
80 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1} $$ [1] $y(0) = 0$; $t_{0}=0$; $\alpha$, $\...
7
votes
0answers
458 views

Convergence/Divergence of infinite series $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\left|{\cos n}\right|}}$

It is well known that $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent while $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon}}$ is convergent for a fixed positive value of $\...
7
votes
0answers
727 views

Gauss-Lucas Theorem (roots of derivatives)

Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots $\alpha_1,\ldots,...
7
votes
0answers
160 views

Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

While working on mixture (variance) of normal distribution and keep running into these two integrals $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ $...
6
votes
0answers
186 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in ...
6
votes
0answers
108 views

Help with the following summation when $x^{37}=1,x\neq 1$

I want to find the following summation $\text{Let }x^{37} = 1 \text{ and } x \neq 1,$ $\\ \text{Find the summation of }$ $$\frac{1}{(1+x+x^2+x^3)^3}+\frac{2}{(1+x^2+x^4+x^6)^3}+...+\frac{36}{(1+x^{...
6
votes
0answers
417 views

Fourier transform of integral related to zeta function

In this MO question here, I asked about the Fourier transform of the zeta function. The second answer lists the following as a representation for $\zeta(s)$, with $E(x)$ as the floor function: \begin{...
6
votes
0answers
138 views

How do people pick $\delta$ so fast in $\epsilon-\delta$ proofs

For example, in a proof that shows $f(x) = \sqrt(x)$ is uniformly continuous on the positive real line, the proof goes like: Let $\epsilon > 0$ be given, and $\delta = \epsilon^2$.... Or to ...
6
votes
0answers
91 views

How should I calculate the $n$th derivative of this expression?

What would be the $n$th derivative of $ f (x) = x ^ x$ I have reached the fifth derivative, very long indeed but I see no pattern that will help me find a general expression. 1 D $y=x^x / ln ...
6
votes
0answers
167 views

Freshman calculus - Stokes's theorem proof

Many calculus text books and courses do not introduce full proof of Stokes's theorem because of differential forms and topological concepts. There are only restrict proofs (for example, simple ...