A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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5
votes
3answers
110 views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
0
votes
0answers
26 views

Closed form of integral involving Bessel J_0

I am trying to find a closed form (in terms of known functions) for this integral: $$ \int_{0}^\Lambda\!\!\! \text{d} k\, J_0( k x)\sin (k y)$$ where $x>0$, $\Lambda>0$, $y\in \mathbb{R}$ I ...
6
votes
2answers
129 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
5
votes
0answers
82 views

Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$

(This is a slight variation of another question, already answered) Can we find a closed form of the following series? $$S=\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}\tag1$$ Using some non-rigorous ...
0
votes
0answers
27 views

Minimum of sum of squares over sums

I am trying to minimize $\phi(\alpha)$, where $\alpha \in \mathbb{R}^K$. $\phi(\alpha) = \frac{R^2 + G^2 \gamma \sum_{i=0}^{K} A_i \alpha_i^2}{\sum_{i=0}^{K} A_i \alpha_i} $ Where, $A_i = \gamma ...
5
votes
2answers
534 views

Only once differentiable

Is there any example of a real function that is one-time-only differentiable, meaning there is $f'(x)$, but no $f''(x)$? I haven't been able to find any example... Of course it would be preferred if f ...
1
vote
2answers
152 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
6
votes
3answers
120 views

Closed-form of $\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right)$

I've found the following identity while I was going through a quite difficult path. $$ \Re\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right) = \frac{\pi^2}{24} -\frac{1}{2}\ln^2 2 - ...
13
votes
4answers
241 views

A conjectured result for $\sum_{n=1}^\infty\frac{(-1)^n\,H_{n/5}}n$

Let $H_q$ denote harmonic numbers (generalized to a non-integer index $q$): $$H_q=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+q}\right)=\int_0^1\frac{1-x^q}{1-x}dx=\gamma+\psi(q+1),\tag1$$ where ...
4
votes
2answers
68 views

Does the elliptic function $\operatorname{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\big|\frac{1}{2}\right)$ have a closed form?

Given the complete elliptic integral of the first kind $K(k)$ for the modulus $k$, can the elliptic function $$\text{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\bigg|\frac{1}{2}\right)$$ be ...
4
votes
1answer
61 views

closed-form of an integral similar to Bessel function

The integral form of the $n$-th modified Bessel function of the first kind is $$ I_n(z)=\frac{1}{\pi}\int_0^{\pi}e^{z\cos\theta}\cos(n\theta)\;d\theta. $$ However, I found an integral $$ ...
28
votes
1answer
394 views

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln ...
6
votes
2answers
187 views
4
votes
3answers
142 views

Closed-form of $\int_0^1 \operatorname{Li}_3\left(1-x^2\right) dx$

By using dilogarithm functional equations we can show that $$ \int_0^1 \operatorname{Li}_2\left(1-x^2\right)\,dx = \frac{\pi^2}{2}-4, $$ where $\operatorname{Li}_2$ is the dilogarithm function. Could ...
2
votes
1answer
53 views

Computing the integral of $-1/f''$

I think this is a very silly question but I have some problems nonetheless. If I know that $g'=-\frac{1}{f''}$, is then $$ g=(f')^{-1}? $$
3
votes
1answer
105 views

Hypergeometric function values and the Baxter constant

While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant: ...
1
vote
0answers
47 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
8
votes
4answers
174 views

Is there a formulaic way to go from $\sum_{k=1}^{n} \frac{1}{k}$ back to $n$?

Say you want to sum $g(n) = \sum_{k=1}^{n} \frac{1}{k} = L$. Is there a simple formula to go from $L$ and deduce $n$? My attempt: For $n = 1$, the formula is $L$. Assume there is a formula for all ...
6
votes
1answer
143 views

Closed-form of the hypergeometric function ${_4F_3}\left(\begin{array}c1,1,\tfrac54,\tfrac74\\\tfrac32,2,2\end{array}\middle|\,-t\right)$

Inspired by this question and by using Mathematica the following conjecture seems to be true for all nonzero complex $t$ number: ...
8
votes
1answer
88 views

$\sum_{n=1}^\infty \frac{1}{(n^2-1)!} - \sum_{n=1}^\infty \frac{1}{(7n+1)!}$ is almost $1+1/6$

I've recognized, that $$\mathcal{S} = \sum_{n=1}^\infty \frac{1}{(n^2-1)!} - \sum_{n=1}^\infty \frac{1}{(7n+1)!} \approx 1.1666666666666666666657785992648796$$ which is almost $1+1/6$. I think it is ...
5
votes
1answer
197 views

Integral $\int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx$

I've conjectured the following identity for $n\geq0$ integers: $$ \int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx = (-1)^{n+1}n! \cdot \left(-\operatorname{Ei}(1)+\sum_{k=1}^{n+1} ...
10
votes
1answer
147 views
+50

This 1 innocent looking recurrence relation seems to have no closed form solution.

$$P(c \cdot x) = \cos(x) P(x)$$ For $c=2$, $P(x) = \sin(x)/x$ is a solution to this. I don't know if there's a closed-form solution for $c \ne 2$. Rather than add my own attempt at solution, which ...
-1
votes
1answer
55 views

Transcendental equation $2 x n\cot (2x)= x^2 - n^2$

I have a transcendental equation and I have not a mathematical superiour formation (I'm an hydraulic engineer) necessary to solve it. The equation is : $2 x n\cot (2x)= x^2 - n^2$ or (same equation) ...
1
vote
1answer
31 views

How to prove that a sequence with recurrence relation has no closed form expression?

It is always easy to forge recurrence relations. E.g. $$a_{n+1}=2a_n+\dfrac{1}{a_n}, a_0=1$$ But it is always hard to find the general closed form expression. And it is even harder to prove that ...
7
votes
1answer
158 views

Evaluating $\sum_{n \geq 1}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)$

Is there a direct way to evaluate the following series? $$ \sum_{n=1}^{\infty}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)=\frac12\ln^2 2. \tag1 $$ I've tried telescoping ...
3
votes
0answers
78 views

Infinite Series $\sum_{m=1}^{\infty} \frac{(-1)^{m-1}}{m^m}$

Any ideas to calculate this infinite sum? The ratio test guarantees the convergence; $$\lim_{m\to \infty} \frac{m^m}{(m+1)^{(m+1)}}=0<1$$
5
votes
2answers
145 views

Closed form for $(2^1-1)(2^2-1)…(2^k-1)$?

Is there closed form for $\prod_1^{i=k}(2^i-1)$ ? I found that it is the product of the terms of the following arithmetico-geometric sequence : $$\{u_1=1,u_{n+1}=2u_n+1\}$$ I found nothing with ...
1
vote
3answers
97 views

A nifty series involving $\cosh(x)$

There are many series that can be found in the literature that are entertaining. Here is yet another. What is the resulting value of the series $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \, ...
4
votes
0answers
107 views

closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
8
votes
2answers
128 views

Proving $~\prod~\frac{\cosh\left(n^2+n+\frac12\right)+i\sinh\left(n+\frac12\right)}{\cosh\left(n^2+n+\frac12\right)-i\sinh\left(n+\frac12\right)}~=~i$

How could we prove that $${\LARGE\prod_{\Large n\ge0}}~\frac{\cosh\left(n^2+n+\dfrac12\right)+i\sinh\left(n+\dfrac12\right)}{\cosh\left(n^2+n+\dfrac12\right)-i\sinh\left(n+\dfrac12\right)}~=~i$$ ...
5
votes
0answers
302 views

Challenging integral: $\int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx$

I'd like to find a symbolic form for the following integral: $$ f(\alpha, Z) = \int_0^Z\frac{\alpha^{(1-x^2)}}{1-x^2} dx $$ It is given that $0 \le \alpha \le 1$ and $0 \le Z < 1$. The following ...
2
votes
3answers
182 views

Generating Functions and closed form [closed]

I read somewhere that we can use generating functions to find closed form of a sequence. So what is the difference between a generating function and closed form of a sqeunce?
13
votes
5answers
448 views

Closed form for an infinite product

The following fascinating formula appears in the paper "On gamma quotients and infinite products" by M.Chamberland and A.Straub (see page 9): ...
0
votes
0answers
16 views

Calculate in closed form the number of questions to gamble to maximize the probability to get a given score in a multiple choice test

I am trying to generalize and calculate in closed form the best strategy that a gambler should follow to maximize the probability to get a given score in a multiple choice test (i.e. the number of ...
10
votes
4answers
370 views

explicit formula for recurrence relation $a_{n+1}=2a_n+\frac{1}{a_n}$

For $n\in\mathbb N$, $$a_{n+1}=2a_n+\frac{1}{a_n},\quad a_1=1. $$ Can any one give an explicit formula for all $a_n$? If such an explicit general formula doesn't exist, please explain it. I've tried ...
6
votes
3answers
106 views

Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$

I've conjectured, that for $n\geq0$ and $m\geq1$ integers $$ \int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right), $$ where $\operatorname{li}$ is the ...
10
votes
1answer
84 views

Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$

Let denote $K$ and $E$ the complete elliptic integral of the first and second kind. The integrand $K(\sqrt{k})$ and $E(\sqrt{k})$ has a closed-form antiderivative in term of $K(\sqrt{k})$ and ...
13
votes
2answers
206 views

Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$

Please help me to evaluate this integral in a closed form: $$I=\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$$ Using integration by parts I found that it could be expressed through ...
0
votes
0answers
58 views

Closed form solution (formula) for possible events

Let's have 100 time units and 4 possible events A1, A2, B1, B2 that might occur within the 100 units. A1 always occurs before A2, B1 always occurs before B2, t1 < t2 < t3 < t4. There are 2 ...
2
votes
1answer
83 views

What is $\int_0^{\pi} \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} \, dx$?

I read this question. The integral has a special property, so it might possibly be evaluable? No one tried evaluating it, so I created this. Not very often I ask question like this, but here it is. ...
0
votes
1answer
19 views

Simple closed functional form for summed recurrence relation

I'm struggling to obtain a simple closed form for a summed recurrence relation. I have an overall form $y=A\left(n-\sum_i^n\frac{e^{-x_i}}{B}\right)$ where $x_{i+1}=kx_i+m$ with $A,B,m >1$ and ...
0
votes
1answer
47 views

A formula to calculate the partial volume of a capsule or tank?

We are trying to ascertain the correct formula discussed in this post. The volume formula for a capsule (a cylinder with a hemisphere at both ends) is, $$V_c = \pi r^2 H + \frac{4}{3}\pi r^3\tag1$$ ...
7
votes
2answers
205 views

Closed-form of $\int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

I've conjectured the following closed-form: $$ I = \int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx = -2\,G\,\ln2, $$ where $G$ is ...
2
votes
0answers
71 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting: $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
8
votes
2answers
195 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
6
votes
2answers
131 views

The value of the integral $\int_0^2\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$

The value of definite integral $$\int\limits_{0}^{2}\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$$ is $$(A)\,4 \quad(B)\,5 \quad (C)\,6 \quad(D)\,7$$ My attempt: I tried using ...
2
votes
0answers
58 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
1
vote
1answer
198 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
1
vote
1answer
30 views

Summation of infinite series, where difference in consecutive denominator forms an A.P.

What is the sum of an infinite series where each term can be written as $\frac{p}{q}$, where p=1 always the difference between 2 consecutive denominators forms an A.P. For example $\dfrac{1}{2}$, ...
4
votes
2answers
114 views

Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?

I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to $$ \sum_{n=0}^{\infty} \frac{1}{1+a^n} $$ is there a closed form with a name? I suspect ...