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

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70
votes
0answers
2k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
27
votes
0answers
524 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 ...
18
votes
0answers
370 views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method which can find this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
15
votes
0answers
247 views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get ...
13
votes
0answers
93 views

A combination integral and series resulting the inverse tangent integral

$\def\Ti{{\rm{Ti}}_2}$I have been able to solve an integral problem, now I tried to use the other method to crack the integral and I have to prove the following expression \begin{equation} ...
13
votes
0answers
224 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 ...
12
votes
0answers
143 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 ...
12
votes
0answers
263 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
12
votes
0answers
231 views

Evaluting $\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} \operatorname dx$

$$\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx$$ My try:: $\displaystyle \int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx = ...
11
votes
0answers
160 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
10
votes
0answers
172 views

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

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 $$ ...
9
votes
0answers
170 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. $$ ...
9
votes
0answers
322 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 $$ ...
9
votes
0answers
86 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
335 views

The same bit of trivial algebra in two different places?

The Villarceau circles are things whose existence is surprising. To find radii of Villarceau circles, I stupidly went through a bit of trigonometry and got a much simpler result than I expected, and ...
8
votes
0answers
60 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 ...
8
votes
0answers
111 views

Evaluating the double limit $\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2m}(n! \pi x)$

I have to find out the following limit $$\lim_{m\to\infty}\lim_{n\to\infty}[\cos(n!πx)^{2m}]$$ for $x$ rational and irrational. for $x$ rational $x$ can be written as $\frac{p}{q}$ and as $n!$ will ...
8
votes
0answers
117 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.
8
votes
0answers
165 views

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

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 ...
8
votes
0answers
208 views

Definite integral $\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$

Could you help me finding the following definite integral, with $a$ and $b$ constants? Thank you! $$\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$$
8
votes
0answers
161 views

Algorithm for obtaining the surface of a mirror

My colleague and I have been trying to implement an algorithm described in the paper "Recovering local shape of a mirror surface from reflection of a regular grid", primary author of which being ...
8
votes
0answers
162 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 ...
7
votes
0answers
55 views

Hard sum with harmonics numbers

Prove or disprove that $S=\displaystyle\sum_{n=1}^{\infty}\frac{{H_n^{2}}~{H_n^{(2)}}+3{H_n^{(4)}}}{n~2^n}=\frac{25}{16}\zeta(5)+\frac{7}{8}\zeta(2)\zeta(3)$.
7
votes
0answers
120 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
7
votes
0answers
104 views

Length of parametric curve $\phi(t)=(f(t)\cos(t),f(t)\sin(t))$

Define the curve $\phi$ by $\phi(t):=(f(t)\cos(t),f(t)\sin(t))$, where $f$ be a strictly increasing infinitly many differentiable function . Find an explicit formula for the length of $\phi$ ...
7
votes
0answers
104 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
113 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
347 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
193 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
109 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
89 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} ...
7
votes
0answers
180 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 ...
7
votes
0answers
290 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
120 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
185 views

Limit of a product of trigonometric functions

I have to find the following limit: $$J=\lim_{N\to\infty}\prod_{k=1}^N \cos\left(\frac{\pi}{k^2}\right).$$ How can I calculate it? Thanks
7
votes
0answers
125 views

Other methods for Laplacian equations

Assume $$A^{2}=(x^{2}+y^{2})\cos^{2}\psi+z^{2}\cot^{2}\psi$$ which $A$ is constant. How we can show $\psi(x,y,z)$ satisfies the Laplacian equation $\psi_{xx}+\psi_{yy}+\psi_{zz}=0$ ...
6
votes
0answers
139 views

Closed-form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
6
votes
0answers
88 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 ...
6
votes
0answers
115 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
6
votes
0answers
63 views

Symbolic manipulation inside integral

I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic ...
6
votes
0answers
132 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
6
votes
0answers
104 views

general solution of the equation $\frac{dy}{dx} =\exp(y/x)$

How can i get the general solution of the equation a) $\frac{dy}{dx} = \exp(y/x)$ b) $\frac{dy}{dx} = \exp(x-y)$ and $y=2$ when $x = 0$ I tried b) first: This is a first-order nonlinear ordinary ...
6
votes
0answers
146 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
99 views

Doubt about integration over a curve

I'm reading a paper and I have a doubt: Suppose we have a compact Riemannian manifold and an eigenfunction of the Laplace operator $\Delta u=-\lambda u$. Suppose in addition that we have the following ...
6
votes
0answers
100 views

What is wrong with my contradiction?

Spivak says the following function does not have an integral: $$ F(x) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q}\\ 0 & : x \notin \mathbb{Q} \end{array} ...
6
votes
0answers
67 views

If all convex combinations of $p(x)$ and $q(x)$ have real roots, then $p,q$ have a common interlacing poly

I heard this result in a talk the other day: Suppose $p$ and $q$ are polynomials. Suppose $p$ is a polynomial of degree $n$ and $q$ a polynomial of degree $n-1$. Call $q$ and interlacer of $p$ if the ...
6
votes
0answers
164 views

Integral mean value theorem- relate a point in the derivative to an integral

Question: Let $f:[a,b] \to \Bbb R$ be a continuously differentiable function s.t $f(a)=f(b)=0$ Prove that exists a point $c \in (a,b)$ such that $$ |f'(c)| \ge \frac 4{(b-a)^2} \int ^b_a f(x) dx $$ ...
6
votes
0answers
144 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
377 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 ...
6
votes
0answers
145 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 $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ ...